Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems

1. Introduction

It is known that if a Mersenne number of the form $M_n=(2^n-1)$ is prime, then n is prime. The reciprocal is not true, as for example for $n=11$, $M_{11}$ is composite, $M_{11}=2047=23\ast 89$ (for review, see e.g., [4,23,34]). There are 51 Mersenne prime numbers known [13]. The largest appears for $n=82589933$, $M_{82589933}=\left(2^{82589933}-1\right)$, and has 24862048 digits.

Due to their intensive use in cryptography, several generalizations of Mersenne numbers have been proposed, first by Crandall [8] of the form $(2^n-C)$ where C is a small odd natural integer number; then by Solinas [29,30,31] of the form $\left(2^n+{ϵ}_3{2}^{m_3}+{ϵ}_2{2}^{m_2}+{ϵ}_1{2}^{m_1}+{ϵ}_0\right)$ which generalized also Fermat numbers and where ${ϵ}_i=-1$, 0 or $+1$, $m_i$ and n are multiple of s, the length of a computer word (e.g. $s=32$); and finally further generalized [10] in the form $\left(2^n+{\sum }_{i=1}^k\left[{ϵ}_i{2}^{m_i}\right]+{ϵ}_0\right)$ with n, k and $m_i$ natural integers, $1\le k<n$, $1\le m_i<n$ and ${ϵ}_i=-1$, 0 or $+1$. Hoque and Saikia proposed [16,17] another definition of generalized Mersenne numbers as $M_{p,q}=(p^q-p+1)$ where $p,q$ are positive integers. Deng introduced [11] a different definition of generalized Mersenne primes which is of the form $R(k,p)=(p^k-1)/(p-1)$, where $k,p$ and $R(k,p)$ are prime numbers.

In this paper, we propose another generalized definition of Mersenne numbers of the form $\left(a^n-\left(a-1\right){ }^n\right)$ with a and n natural integers. Although the name Generalized Mersenne number is already in use for pseudo-Mersenne numbers of the form proposed by Crandall [8], Solinas [29,30,31], and others, we propose to call them Global Generalized Mersenne numbers, or in short Generalized Mersenne ($G{M}_{a,n}$) numbers, referring to the fact that both the base a and the exponent n can take any integer values $>1$.

Generalized Mersenne numbers are defined in Section 2.1. Section 2.2 gives several decompositions of $G{M}_{a,n}$. Several theorems on congruence of Mersenne numbers are generalized for $G{M}_{a,n}$ in Section 2.3. Congruence properties of $G{M}_{a,n}$ and of their factors are investigated in Section 2.4. The density of Mersenne primes and the potential use of Generalized Mersenne primes in cryptography are shortly discussed in Section 3. Conclusions are drawn in Section 4.

2. Materials and Methods

2.1. Generalized Mersenne numbers

Mersenne numbers can be seen as the difference of the $n^{th}$ power of the first two successive integers

$$M_n=\left(2^n-1\right)=\left(2^n-1^n\right)$$

(2.1)

By extension, Generalized Mersenne (GM) numbers, noted $G{M}_{a,n}$, are defined as the difference of the $n^{th}$ power of two successive integers

$$G{M}_{a,n}=\left(a^n-\left(a-1\right){ }^n\right)$$

(2.2)

and indexed by the base a and the exponent n, with $a\ge 2$ and $n\ge 2$ natural integers.

It is easy to show, like for Mersenne numbers, that Generalized Mersenne numbers can only be primes if n itself is prime. Indeed, if n is composite, $n=rs$ with r and s natural positive integers, then all $G{M}_{a,n}=\left(a^{rs}-\left(a-1\right){ }^{rs}\right)$ are binomial numbers, having $\left(a^r-\left(a-1\right){ }^r\right)$ or $\left(a^s-\left(a-1\right){ }^s\right)$ as integer factor. Therefore, in the rest of this paper, we will consider only the cases of n being prime.

Table 1 shows the first 25 $G{M}_{a,n}$ numbers for the first five primes $n=2,3,5,7,11$, with $G{M}_{a,n}$ prime and composite numbers shown respectively in bold and italic characters.

For $n=2$, (2.2) yields all the odd integers $G{M}_{a,2}=2a-1$. For $n=3$, the first four $G{M}_{a,3}$ numbers are prime for $a=2$ to 5; further numbers are composite or prime without any seemingly regular pattern. For $n=5$ and 7 and $a=2$, $G{M}_{2,5}$ and $G{M}_{2,7}$ are the Mersenne primes $M_5$ and $M_7$. For $3\le a\le 19$, interesting patterns occur in the two $G{M}_{a,5}$ and $G{M}_{a,7}$ series. For $a=3$ and 4, $G{M}_{a,5}$ and $G{M}_{a,7}$ are oppositely prime and composite. For $a=5$, $G{M}_{a,5}$ and $G{M}_{a,7}$ are both composites. For $a=6$ to 12, $G{M}_{a,5}$ and $G{M}_{a,7}$ are oppositely primes and composites again, with a series of composite $G{M}_{a,5}$ and prime $G{M}_{a,7}$ for $a=7$ to 10. For $a=13$ to 19, $G{M}_{a,5}$ and $G{M}_{a,7}$ are composites or primes for same values of a. For larger values of a, regular patterns between $G{M}_{a,5}$ and $G{M}_{a,7}$ disappear and reappear for certain ranges of values of a. For $n=11$, the first four $G{M}_{a,11}$ are composite (the fifth Mersenne number $M_{11}=2047$ is not prime). Among the first 25 $G{M}_{a,11}$, the values for $a=6$, 8, 10 and 14 yield prime numbers.

It is observed that, for odd values of n with $n\equiv 1\left(mod4\right)$, the series of $G{M}_{a,n}$ numbers generated for successive values of the base a have 1 as last digit, while for odd values of n with $n\equiv 3\left(mod4\right)$, the series of the last digit of $G{M}_{a,n}$ numbers are repetitions of the sequence 1, 7, 9, 7, 1 respectively for bases $a\equiv k\left(mod5\right)$, with k respectively 1, 2, 3, 4, 0. This is demonstrated further in Section 2.3.3.

The cause of these patterns, or lack of it, in the distributions of composite and prime generalized Mersenne numbers is tantalizing. The beginning of an answer is given in the next sections.

Table 1. First 25 $G{M}_{a,n}$ numbers for $n=2$, 3, 5, 7, 11.

a	n = 2	n = 3	n = 5	n = 7	n = 11
2	3	7	31	127	2047
3	5	19	211	2059	175099
4	7	37	781	14197	4017157
5	9	61	2101	61741	44633821
6	11	91	4651	201811	313968931
7	13	127	9031	543607	1614529687
8	15	169	15961	1273609	6612607849
9	17	217	26281	2685817	22791125017
10	19	271	40951	5217031	68618940391
11	21	331	61051	9487171	185311670611
12	23	397	87781	16344637	457696700077
13	25	469	122461	26916709	1049152023349
14	27	547	166531	42664987	2257404775627
15	29	631	221551	65445871	4600190689711
16	31	721	289201	97576081	8942430185041
17	33	817	371281	141903217	16679710263217
18	35	919	469711	201881359	29996513771599
19	37	1027	586531	281651707	52221848818987
20	39	1141	723901	386128261	88309741101781
21	41	1261	884101	521088541	145477500542221
22	43	1387	1069531	693269347	234040800869107
23	45	1519	1282711	910467559	368491456502599
24	47	1657	1526281	1181645977	568871385255097
25	49	1801	1803001	1517044201	862504647846601

Table 1. First 25 $G{M}_{a,n}$ numbers for $n=2$, 3, 5, 7, 11.

a	n = 2	n = 3	n = 5	n = 7	n = 11
2	3	7	31	127	2047
3	5	19	211	2059	175099
4	7	37	781	14197	4017157
5	9	61	2101	61741	44633821
6	11	91	4651	201811	313968931
7	13	127	9031	543607	1614529687
8	15	169	15961	1273609	6612607849
9	17	217	26281	2685817	22791125017
10	19	271	40951	5217031	68618940391
11	21	331	61051	9487171	185311670611
12	23	397	87781	16344637	457696700077
13	25	469	122461	26916709	1049152023349
14	27	547	166531	42664987	2257404775627
15	29	631	221551	65445871	4600190689711
16	31	721	289201	97576081	8942430185041
17	33	817	371281	141903217	16679710263217
18	35	919	469711	201881359	29996513771599
19	37	1027	586531	281651707	52221848818987
20	39	1141	723901	386128261	88309741101781
21	41	1261	884101	521088541	145477500542221
22	43	1387	1069531	693269347	234040800869107
23	45	1519	1282711	910467559	368491456502599
24	47	1657	1526281	1181645977	568871385255097
25	49	1801	1803001	1517044201	862504647846601

2.2. Decomposition of Generalized Mersenne Numbers

It is known that all Mersenne numbers and their factors can be written in the form

$$M_n=2nq+1$$

(2.3)

with q and n positive natural integer and n prime (see e.g. [23,27] and [6]). All Generalized Mersenne numbers can also be written in a similar form as demonstrated in the following theorem.

Theorem 1.

For a and n natural integers, $n>2$, all Generalized Mersenne numbers can be written as

$$G{M}_{a,n}=2n{Q}_n\left(a\right)+1$$

(2.4)

for all prime exponents $n>2$ and for all bases a, and where $Q_n\left(a\right)$ is a polynomial in a of degree $n-1$.

Proof. 

Let a, n, i and j be natural integers, with n prime, $n>2$ and $i<n$. Posing

$$d_i^n=\frac{C_i^n}{n}=\frac{\left(n-1\right)!}{i!\left(n-i\right)!}$$

(2.5)

with $C_i^n$ the binomial coefficient, writing Δ for convenience for the triangular number of $\left(a-1\right)$, $▵=▵\left(a-1\right)=\frac{a\left(a-1\right)}{2}$, and noting that the exponent n is odd, developing the polynomial (2.2) yields

$$\begin{array}{cc}\hfill G{M}_{a,n}& =\left(a^n-\left(a^n+\sum _{i=1}^{n-1}\left[{\left(-1\right)}^i{C}_i^n{a}^{n-i}\right]-1\right)\right)=\sum _{i=1}^{n-1}\left[{\left(-1\right)}^{i+1}{C}_i^n{a}^{n-i}\right]+1\hfill \\ \hfill & =n\sum _{i=1}^{n-1}\left[{\left(-1\right)}^{i+1}{d}_i^n{a}^{n-i}\right]+1=n\sum _{i=1}^{\frac{n-1}{2}}\left[{\left(-1\right)}^{i+1}{d}_i^n\left(a^{n-i}-a^i\right)\right]+1\hfill \\ \hfill & =n\sum _{i=1}^{\frac{n-1}{2}}\left[{\left(-1\right)}^{i+1}{d}_i^n{a}^i\left(a^{n-2i}-1\right)\right]+1\hfill \\ \hfill & =n\sum _{i=1}^{\frac{n-1}{2}}\left[{\left(-1\right)}^{i+1}{d}_i^n{a}^i\left(a-1\right)\sum _{j=0}^{n-1-2i}\left[a^{n-1-2i-j}\right]\right]+1\hfill \\ \hfill & =na\left(a-1\right)\sum _{i=1}^{\frac{n-1}{2}}\left[{\left(-1\right)}^{i+1}{d}_i^n\sum _{j=0}^{n-1-2i}\left[a^{n-2-i-j}\right]\right]+1\hfill \\ \hfill & =2n▵\sum _{i=1}^{n-2}\left[S_i^{\left(1\right)}{a}^{n-2-i}\right]+1\hfill \end{array}$$

(2.6)

where, for $1\le i\le \frac{n-1}{2}$,

$$S_i^{\left(1\right)}=\sum _{j=1}^i\left[{\left(-1\right)}^{j+1}{d}_j^n\right]$$

(2.7)

and for $\frac{n+1}{2}\le i\le n-2$,

$$S_i^{\left(1\right)}=\sum _{j=i+1}^{n-1}\left[{\left(-1\right)}^j{d}_j^n\right]=S_{n-1-i}^{\left(1\right)}$$

(2.8)

Relation (2.6) shows that the form (2.4) is obtained where the positive integer function $Q_n\left(a\right)$ is only function of the variable a and is a polynomial in a of degree $n-1$. □

One can characterize further the integer function $Q_n\left(a\right)$ for higher values of n as follows.

Theorem 2.

For a and n natural integers, $n>2$, all Generalized Mersenne numbers can be written as

$$G{M}_{a,n}=2n\left(▵Q_n^{\prime }\left(2▵\right)\right)+1$$

(2.9)

for all prime exponents $n\ge 3$, and as

$$G{M}_{a,n}=2n\left(▵\left(2▵+1\right)Q{"}_n\left(2▵\right)\right)+1$$

(2.10)

for all prime exponents $n\ge 5$ and for all bases a, where $Q_n^{\prime }\left(2▵\right)$ and $Q{"}_n\left(2\Delta \right)$ are polynomials in the variable $▵\left(a-1\right)$ only, the triangular number of $\left(a-1\right)$, and of degrees $\left(\frac{n-3}{2}\right)$ and $\left(\frac{n-5}{2}\right)$ respectively.

Proof. 

Let a, n, i, j, J, k and K be natural integers, with n prime, $n>2$ and $i<n$. Continuing from (2.6) the development of the polynomial (2.2) in $\frac{\left(n-1\right)}{2}$ successive iterations, one obtains an expression of $G{M}_{a,n}$ of the form of either (i) $\frac{\left(n-1\right)}{2}$ embedded products in the new variable $▵=▵\left(a-1\right)$ or (ii) a polynomial of degree $\frac{\left(n-1\right)}{2}$ in ▵.

(i) By arranging similarly the terms $a^{n-2-i}$ in successive differences, a second iteration yields from (2.6)

$$\begin{array}{cc}\hfill G{M}_{a,n}& =2n▵\left(\sum _{i=1}^{n-4}\left[S_i^{\left(2\right)}\left(a^{n-2-i}-a^{n-3-i}\right)\right]+S_{n-2}^{\left(1\right)}\right)+1\hfill \\ \hfill & =2n▵\left(\sum _{i=1}^{n-4}\left[S_i^{\left(2\right)}{a}^{n-3-i}\left(a-1\right)\right]+S_{n-2}^{\left(1\right)}\right)+1\hfill \\ \hfill & =2n▵\left(2▵\sum _{i=1}^{n-4}\left[S_i^{\left(2\right)}{a}^{n-4-i}\right]+1\right)+1\hfill \end{array}$$

(2.11)

where

$$S_i^{\left(2\right)}=\sum _{j=1}^i\left[S_j^{\left(1\right)}\right],S_{n-2}^{\left(1\right)}=S_1^{\left(1\right)}=1$$

(2.12)

Repeating the process of rearranging the terms $a^{n-4-i}$, the third iteration yields from (2.11)

$$G{M}_{a,n}=2n▵\left(2▵\left(2▵\sum _{i=1}^{n-6}\left[S_i^{\left(3\right)}{a}^{n-6-i}\right]+S_{n-4}^{\left(2\right)}\right)+1\right)+1$$

(2.13)

where

$$S_i^{\left(3\right)}=\sum _{j=1}^i\left[S_j^{\left(2\right)}\right],S_{n-4}^{\left(2\right)}=\frac{n-3}{2}$$

(2.14)

that can be easily demonstrated from the equality for odd n

$$\sum _{i=0}^{\frac{n-1}{2}}\left[{\left(-1\right)}^i{C}_i^n\right]=2\sum _{i=0}^{\frac{n-1}{2}}\left[{\left(-1\right)}^{i}i{d}_i^n\right]$$

(2.15)

Further iterations yield successively from (2.13)

4^th iteration:

$$G{M}_{a,n}=2n▵\left(2▵\left(2▵\left(2▵\sum _{i=1}^{n-8}\left[S_i^{\left(4\right)}{a}^{n-8-i}\right]+S_{n-6}^{\left(3\right)}\right)+S_{n-4}^{\left(2\right)}\right)+1\right)+1$$

(2.16)

..., $k^{th}$ iteration (for $k\ge 3$):

$$G{M}_{a,n}=2n▵\left(2▵\left(2▵\left(2▵\left(...\left(2▵\sum _{i=1}^{n-2k}\left[S_i^{\left(k\right)}{a}^{n-2k-i}\right]\right.\right.\right.\right.\right.$$

$$\left.\left.\left.\left.\left.+S_{n-2\left(k-1\right)}^{\left(k-1\right)}\right)...\right)+S_{n-6}^{\left(3\right)}\right)+S_{n-4}^{\left(2\right)}\right)+1\right)+1$$

(2.17)

where

$$S_i^{\left(4\right)}=\sum _{j=1}^i\left[S_j^{\left(3\right)}\right],S_i^{\left(k\right)}=\sum _{j=1}^i\left[S_j^{\left(k-1\right)}\right]$$

(2.18)

The iteration process stops after $\frac{\left(n-1\right)}{2}$ iterations yielding finally the $\frac{\left(n-1\right)}{2}$ embedded products

$$G{M}_{a,n}=2n▵\left(2▵\left(2▵\left(2▵\left(...\left(2▵+S_3^{\left(\frac{n-3}{2}\right)}\right)...\right)+S_{n-6}^{\left(3\right)}\right)+S_{n-4}^{\left(2\right)}\right)+1\right)+1$$

(2.19)

The independent terms are

$$\begin{array}{cc}\hfill S_{n-2k}^{\left(k\right)}& =\sum _{j=1}^{n-2k}\left[{\left(-1\right)}^{j+1}\frac{{\prod }_{i=0}^{k-2}\left[n-2k+1-j+i\right]}{\left(k-1\right)!}{d}_j^n\right]\hfill \\ \hfill & =\sum _{j=1}^{n-2k}\left[{\left(-1\right)}^{j+1}{C}_{n-2k-j}^{n-\left(k+1\right)-j}{d}_j^n\right]\hfill \end{array}$$

(2.20)

that can be transformed by a change of index $J=n-2k-j$ in

$$\begin{array}{cc}\hfill S_{n-2k}^{\left(k\right)}& =\sum _{J=0}^{n-2k-1}\left[{\left(-1\right)}^{n-2k+1-J}{C}_J^{k-1+J}\frac{C_{n-2k-J}^n}{n}\right]\hfill \\ \hfill & =\left(\frac{1}{n}\right)\left(\sum _{J=0}^{n-2k}\left[{\left(-1\right)}^J{C}_J^{k-1+J}{C}_{n-2k-J}^n\right]+C_{n-2k}^{n-k-1}\right)\hfill \\ \hfill & =\left(\frac{1}{n}\right)\left(\sum _{J=0}^{n-2k}\left[C_J^{-k}{C}_{n-2k-J}^n\right]+C_{n-2k}^{n-k-1}\right)\hfill \\ \hfill & =\left(\frac{C_{n-2k}^{n-k}+C_{n-2k}^{n-k-1}}{n}\right)=\frac{C_{k-1}^{n-k-1}}{k}=\frac{\left(n-\left(k+1\right)\right)!}{k!\left(n-2k\right)!}\hfill \end{array}$$

(2.21)

and where Vandermonde’s convolution was used, yielding $S_{n-2}^{\left(1\right)}=1$, $S_{n-4}^{\left(2\right)}=\frac{n-3}{2!}$, $S_{n-6}^{\left(3\right)}=\frac{\left(n-4\right)\left(n-5\right)}{3!}$, $S_{n-8}^{\left(4\right)}=\frac{\left(n-5\right)\left(n-6\right)\left(n-7\right)}{4!}$, ... The last independent term (the one in the middle of (2.19)) is found for $k=\frac{n-3}{2}$, yielding

$$S_3^{\left(\frac{n-3}{2}\right)}=\frac{C_{\frac{n-5}{2}}^{\frac{n+1}{2}}}{\frac{n-3}{2}}=\frac{n^2-1}{24}=\frac{n^2-1}{2^2\times 3!}$$

(2.22)

The independent terms before the last one can also be written as

$$S_5^{\left(\frac{n-5}{2}\right)}=\frac{\left(n^2-1\right)\left(n^2-3^2\right)}{2^4\times 5!},S_7^{\left(\frac{n-7}{2}\right)}=\frac{\left(n^2-1\right)\left(n^2-3^2\right)\left(n^2-5^2\right)}{2^6\times 7!},...$$

(2.23)

or more generally for $K=3$ to $\left(n-2\right)$

$$S_K^{\left(\frac{n-K}{2}\right)}=\frac{{\prod }_{i=1}^{\frac{K-1}{2}}\left[n^2-{\left(2i-1\right)}^2\right]}{2^{K-1}\times K!}$$

(2.24)

(ii) Instead of embedded products, a polynomial expression can be found from (2.19) in the form

$$G{M}_{a,n}=2n▵\left(\sum _{i=0}^{\frac{n-3}{2}}\left[{\left(2▵\right)}^i{S}_{n-2\left(i+1\right)}^{\left(i+1\right)}\right]\right)+1=2n▵\left(\sum _{i=0}^{\frac{n-3}{2}}\left[{\left(2▵\right)}^i\frac{C_i^{n-i-2}}{i+1}\right]\right)+1$$

(2.25)

The positive integer function $Q_n\left(a\right)$ in (2.4) can be deduced as a function of Δ either from (2.19)

$$Q_n\left(a\right)=▵\left(2▵\left(2▵\left(2▵\left(...\left(2▵+S_3^{\left(\frac{n-3}{2}\right)}\right)...\right)+S_{n-6}^{\left(3\right)}\right)+S_{n-4}^{\left(2\right)}\right)+1\right)$$

(2.26)

or from (2.25)

$$Q_n\left(a\right)=▵\left(\sum _{i=0}^{\frac{n-3}{2}}\left[{\left(2▵\right)}^i{S}_{n-2\left(i+1\right)}^{\left(i+1\right)}\right]\right)$$

(2.27)

The positive integer function $Q_n^{\prime }\left(2▵\right)$ in (2.9) can be deduced from (2.26) or (2.27) with (2.21)

$$Q_n^{\prime }\left(2▵\right)=\sum _{k=1}^{\frac{n-1}{2}}\left[{\left(2▵\right)}^{k-1}{S}_{n-2k}^{\left(k\right)}\right]=\sum _{k=1}^{\frac{n-1}{2}}\left[{\left(2▵\right)}^{k-1}\frac{C_{k-1}^{n-k-1}}{k}\right]$$

(2.28)

For $n\ge 5$, factoring the right side of (2.25) by $\left(2▵+1\right)$ yields

$$\begin{array}{cc}\hfill G{M}_{a,n}& =2n▵\left(2▵+1\right)\left(\sum _{i=0}^{\frac{n-5}{2}}\left[{\left(2▵\right)}^i\sum _{j=0}^{\frac{n-5}{2}-i}\left[{\left(-1\right)}^{\frac{n-5}{2}-i+j}{S}_{2j+1}^{\left(\frac{n-1}{2}-j\right)}\right]\right]\right)+1\hfill \\ \hfill & =2n▵\left(2▵+1\right)\left(\sum _{i=0}^{\frac{n-5}{2}}\left[{\left(2▵\right)}^i\sum _{j=0}^{\frac{n-5}{2}-i}\left[{\left(-1\right)}^{\frac{n-5}{2}-i+j}\frac{C_{2j}^{\left(\frac{n-1}{2}+j\right)}}{\left(2j+1\right)}\right]\right]\right)+1\hfill \end{array}$$

(2.29)

The positive integer function $Q{"}_n\left(2▵\right)$ in (2.10) is deduced further from (2.29)

$$Q{"}_n\left(2▵\right)=\sum _{i=0}^{\frac{n-5}{2}}\left[{\left(2▵\right)}^i\sum _{j=0}^{\frac{n-5}{2}-i}\left[{\left(-1\right)}^{\frac{n-5}{2}-i+j}{S}_{2j+1}^{\left(\frac{n-1}{2}-j\right)}\right]\right]$$

(2.30)

or inversely, by inverting the sums,

$$Q{"}_n\left(2▵\right)={\left(-1\right)}^{\frac{n-5}{2}}\sum _{j=0}^{\frac{n-5}{2}}\left[{\left(-1\right)}^j\frac{C_{2j}^{\left(\frac{n-1}{2}+j\right)}}{\left(2j+1\right)}\sum _{i=0}^{\frac{n-5}{2}-j}\left[{\left(-2▵\right)}^i\right]\right]$$

(2.31)

Therefore, the general form of all $G{M}_{a,n}$ can be written as in (2.9) and (2.10) for n prime, respectively $n\ge 3$ and $n\ge 5$, where the positive integer functions $Q_n^{\prime }\left(2▵\right)$ and $Q{"}_n\left(2▵\right)$ are polynomials of only the triangular number $▵\left(a-1\right)$ as variable and of degrees respectively $\left(\frac{n-3}{2}\right)$ and $\left(\frac{n-5}{2}\right)$. □

Note that for large values of the exponent n, the calculation of $G{M}_{a,n}$ becomes quickly intractable as $n^{th}$ powers become difficult to compute. The development given in Theorem 2 for odd prime values of n alleviates the problem by reducing the degree of the polynomial (2.2) from n to $\left(\frac{n-1}{2}\right)$.

For very large values of a and n, the value of a $G{M}_{a,n}$ is dominated by the first term in the polynomial (2.25) and can therefore be approximated by

$$G{M}_{a,n}\approx n{a}^{n-1}$$

(2.32)

for $a\gg 1$ and n prime $\gg 1$, the approximation growing better for increasingly larger values of a and n.

For the first six odd prime values of the exponent n, the form (2.19) of embedded products for $G{M}_{a,n}$yields

$$G{M}_{a,3}=2\times 3▵+1$$

(2.33)

$$G{M}_{a,5}=2\times 5▵\left(2▵+1\right)+1$$

(2.34)

$$G{M}_{a,7}=2\times 7▵\left(2▵\left(2▵+2\right)+1\right)+1$$

(2.35)

$$G{M}_{a,11}=2\times 11▵\left(2▵\left(2▵\left(2▵\left(2▵+5\right)+7\right)+4\right)+1\right)+1$$

(2.36)

$$G{M}_{a,13}=2\times 13▵\left(2▵\left(2▵\left(2▵\left(2▵\left(2▵+7\right)+14\right)+12\right)+5\right)+1\right)+1$$

(2.37)

$$G{M}_{a,17}=2\times 17▵\left(2▵\left(2▵\left(2▵\left(2▵\left(2▵\left(2▵\left(2▵+12\right)+42\right)+66\right)+55\right)\right.\right.\right.$$

$$\left.\left.\left.+26\right)+7\right)+1\right)+1$$

(2.38)

while the polynomial expression (2.29) gives, with further factorization,

$$G{M}_{a,7}=2\times 7▵{\left(2▵+1\right)}^2+1$$

(2.39)

$$G{M}_{a,11}=2\times 11▵\left(2▵+1\right)\left[2▵\left(2▵+1\right)\left(2▵+3\right)+1\right]+1$$

(2.40)

$$G{M}_{a,13}=2\times 13▵{\left(2▵+1\right)}^2\left\{\left(2▵+1\right)\left[\left(2▵+1\right)\left(2▵+3\right)-4\right]+2\right\}+1$$

(2.41)

$$G{M}_{a,17}=2\times 17▵\left(2▵+1\right)\left\{\left(2▵+1\right)\left[\left(2▵+1\right)\left(\left(2▵+1\right)\left\{\left(2▵+1\right)\right.\right.\right.\right.$$

$$\left.\left.\left.\left.\left[\left(2▵+1\right)\left(2▵+6\right)-9\right]+1\right\}+6\right)-4\right]+1\right\}+1$$

(2.42)

etc, where, to recall, ▵ is written for $▵\left(a-1\right)$ and where several factorizations are possible for $n\ge 13$. As a further example, Tables 2 show the first ten values of $G{M}_{a,n}$ for prime exponents n from 3 to 11, with the decomposition (2.33), (2.34), (2.39) and (2.40) in integer factors of $\left(G{M}_{a,n}-1\right)$.

Table 2. Decomposition of Generalized Mersenne numbers $G{M}_{a,n}$ for $2\le a\le 10$.

${\mathit{GM}}_{\mathit{a},3}$	$=2\ast 3\ast \mathit{▵}+1$	Decomposition of $\left({\mathit{GM}}_{\mathit{a},3}-1\right)$
7	$=2\ast 3\ast 1+1$	prime
19	$=2\ast 3\ast 3+1$	prime
37	$=2\ast 3\ast 6+1$	prime
61	$=2\ast 3\ast 10+1$	prime
91	$=2\ast 3\ast 15+1$	$=7\ast 13=\left(2\ast 3+1\right)\left(2^2\ast 3+1\right)$
127	$=2\ast 3\ast 21+1$	prime
169	$=2\ast 3\ast 28+1$	$={13}^2={\left(2^2\ast 3+1\right)}^2$
217	$=2\ast 3\ast 36+1$	$=7\ast 31=\left(2\ast 3+1\right)\left(2\ast 3\ast 5+1\right)$
271	$=2\ast 3\ast 45+1$	prime
${\mathit{GM}}_{\mathit{a},\mathbf{5}}$	$=2\ast 5\ast ▵\ast \left(2▵+1\right)+1$	Decomposition of  $\left({\mathit{GM}}_{\mathit{a},\mathbf{5}}-\mathbf{1}\right)$
31	$=2\ast 5\ast 1\ast 3+1$	prime
211	$=2\ast 5\ast 3\ast 7+1$	prime
781	$=2\ast 5\ast 6\ast 13+1$	$=11\ast 71=\left(2\ast 5+1\right)\left(2\ast 5\ast 7+1\right)$
2101	$=2\ast 5\ast 10\ast 21+1$	$=11\ast 191=\left(2\ast 5+1\right)\left(2\ast 5\ast 19+1\right)$
4651	$=2\ast 5\ast 15\ast 31+1$	prime
9031	$=2\ast 5\ast 21\ast 43+1$	$=11\ast 821=\left(2\ast 5+1\right)\left(2^2\ast 5\ast 41+1\right)$
15961	$=2\ast 5\ast 28\ast 57+1$	$=11\ast 1451=\left(2\ast 5+1\right)\left(2\ast 5^2\ast 29+1\right)$
26281	$=2\ast 5\ast 36\ast 73+1$	$=41\ast 641=\left(2^3\ast 5+1\right)\left(2^7\ast 5+1\right)$
40951	$=2\ast 5\ast 45\ast 91+1$	$=31\ast 1321=\left(2\ast 5\ast 3+1\right)\left(2^3\ast 5\ast 3\ast 11+1\right)$
${\mathit{GM}}_{\mathit{a},\mathbf{7}}$	$=2\ast 7\ast ▵\ast {\left(2▵+1\right)}^2+1$	Decomposition of  $\left({\mathit{GM}}_{\mathit{a},\mathbf{7}}-\mathbf{1}\right)$
127	$=2\ast 7\ast 1\ast 3^2+1$	prime
2059	$=2\ast 7\ast 3\ast 7^2+1$	$=29\ast 71=\left(2^2\ast 7+1\right)\left(2\ast 7\ast 5+1\right)$
14197	$=2\ast 7\ast 6\ast {13}^2+1$	prime
61741	$=2\ast 7\ast 10\ast {21}^2+1$	$=29\ast 2129=\left(2^2\ast 7+1\right)\left(2^4\ast 7\ast 19+1\right)$
201811	$=2\ast 7\ast 15\ast {31}^2+1$	$=29\ast 6959=\left(2^2\ast 7+1\right)\left(2\ast 7^2\ast 71+1\right)$
543607	$=2\ast 7\ast 21\ast {43}^2+1$	prime
1273609	$=2\ast 7\ast 28\ast {57}^2+1$	prime
2685817	$=2\ast 7\ast 36\ast {73}^2+1$	prime
5217031	$=2\ast 7\ast 45\ast {91}^2+1$	prime
${\mathit{GM}}_{\mathit{a},\mathbf{11}}$	$=\mathbf{2}\ast \mathbf{11}▵\left(\mathbf{2}▵+\mathbf{1}\right)\left[\mathbf{2}▵\left(\mathbf{2}▵+\mathbf{1}\right)\left(\mathbf{2}▵+\mathbf{3}\right)+\mathbf{1}\right]+\mathbf{1}$	Decomposition of  $\left({\mathit{GM}}_{\mathit{a},\mathbf{11}}-\mathbf{1}\right)$
2047	$=2\ast 11\ast 1\ast 3\left[2\ast 1\ast 3\ast 5+1\right]+1$	$=23\ast 89=\left(2\ast 11+1\right)\left(2^3\ast 11+1\right)$
175099	$=2\ast 11\ast 3\ast 7\left[2\ast 3\ast 7\ast 9+1\right]+1$	$={23}^2\ast 331={\left(2\ast 11+1\right)}^2\left(2\ast 11\ast 3\ast 5+1\right)$
4017157	$=2\ast 11\ast 6\ast 13\left[2\ast 6\ast 13\ast 15+1\right]+1$	$=23\ast 174659=\left(2\ast 11+1\right)\left(2\ast 11\ast 17\ast 467+1\right)$
44633821	$=2\ast 11\ast 10\ast 21\left[2\ast 10\ast 21\ast 23+1\right]+1$	$=6359\ast 7019=(2\ast 11\ast {17}^2+1)(2\ast {11}^2\ast 29+1)$
313968931	$=2\ast 11\ast 15\ast 31\left[2\ast 15\ast 31\ast 33+1\right]+1$	prime
1614529687	$=2\ast 11\ast 21\ast 43\left[2\ast 21\ast 43\ast 45+1\right]+1$	$=89\ast 18140783=(2^3\ast 11+1)(2\ast 11\ast 19\ast 43399+1)$
6612607849	$=2\ast 11\ast 28\ast 57\left[2\ast 28\ast 57\ast 59+1\right]+1$	prime
22791125017	$=2\ast 11\ast 36\ast 73\left[2\ast 36\ast 73\ast 75+1\right]+1$	$=23\ast 990918479=(2\ast 11+1)(2\ast 11\ast 45041749+1)$
68618940391	$=2\ast 11\ast 45\ast 91\left[2\ast 45\ast 91\ast 93+1\right]+1$	prime

Table 2. Decomposition of Generalized Mersenne numbers $G{M}_{a,n}$ for $2\le a\le 10$.

${\mathit{GM}}_{\mathit{a},3}$	$=2\ast 3\ast \mathit{▵}+1$	Decomposition of $\left({\mathit{GM}}_{\mathit{a},3}-1\right)$
7	$=2\ast 3\ast 1+1$	prime
19	$=2\ast 3\ast 3+1$	prime
37	$=2\ast 3\ast 6+1$	prime
61	$=2\ast 3\ast 10+1$	prime
91	$=2\ast 3\ast 15+1$	$=7\ast 13=\left(2\ast 3+1\right)\left(2^2\ast 3+1\right)$
127	$=2\ast 3\ast 21+1$	prime
169	$=2\ast 3\ast 28+1$	$={13}^2={\left(2^2\ast 3+1\right)}^2$
217	$=2\ast 3\ast 36+1$	$=7\ast 31=\left(2\ast 3+1\right)\left(2\ast 3\ast 5+1\right)$
271	$=2\ast 3\ast 45+1$	prime
${\mathit{GM}}_{\mathit{a},\mathbf{5}}$	$=2\ast 5\ast ▵\ast \left(2▵+1\right)+1$	Decomposition of  $\left({\mathit{GM}}_{\mathit{a},\mathbf{5}}-\mathbf{1}\right)$
31	$=2\ast 5\ast 1\ast 3+1$	prime
211	$=2\ast 5\ast 3\ast 7+1$	prime
781	$=2\ast 5\ast 6\ast 13+1$	$=11\ast 71=\left(2\ast 5+1\right)\left(2\ast 5\ast 7+1\right)$
2101	$=2\ast 5\ast 10\ast 21+1$	$=11\ast 191=\left(2\ast 5+1\right)\left(2\ast 5\ast 19+1\right)$
4651	$=2\ast 5\ast 15\ast 31+1$	prime
9031	$=2\ast 5\ast 21\ast 43+1$	$=11\ast 821=\left(2\ast 5+1\right)\left(2^2\ast 5\ast 41+1\right)$
15961	$=2\ast 5\ast 28\ast 57+1$	$=11\ast 1451=\left(2\ast 5+1\right)\left(2\ast 5^2\ast 29+1\right)$
26281	$=2\ast 5\ast 36\ast 73+1$	$=41\ast 641=\left(2^3\ast 5+1\right)\left(2^7\ast 5+1\right)$
40951	$=2\ast 5\ast 45\ast 91+1$	$=31\ast 1321=\left(2\ast 5\ast 3+1\right)\left(2^3\ast 5\ast 3\ast 11+1\right)$
${\mathit{GM}}_{\mathit{a},\mathbf{7}}$	$=2\ast 7\ast ▵\ast {\left(2▵+1\right)}^2+1$	Decomposition of  $\left({\mathit{GM}}_{\mathit{a},\mathbf{7}}-\mathbf{1}\right)$
127	$=2\ast 7\ast 1\ast 3^2+1$	prime
2059	$=2\ast 7\ast 3\ast 7^2+1$	$=29\ast 71=\left(2^2\ast 7+1\right)\left(2\ast 7\ast 5+1\right)$
14197	$=2\ast 7\ast 6\ast {13}^2+1$	prime
61741	$=2\ast 7\ast 10\ast {21}^2+1$	$=29\ast 2129=\left(2^2\ast 7+1\right)\left(2^4\ast 7\ast 19+1\right)$
201811	$=2\ast 7\ast 15\ast {31}^2+1$	$=29\ast 6959=\left(2^2\ast 7+1\right)\left(2\ast 7^2\ast 71+1\right)$
543607	$=2\ast 7\ast 21\ast {43}^2+1$	prime
1273609	$=2\ast 7\ast 28\ast {57}^2+1$	prime
2685817	$=2\ast 7\ast 36\ast {73}^2+1$	prime
5217031	$=2\ast 7\ast 45\ast {91}^2+1$	prime
${\mathit{GM}}_{\mathit{a},\mathbf{11}}$	$=\mathbf{2}\ast \mathbf{11}▵\left(\mathbf{2}▵+\mathbf{1}\right)\left[\mathbf{2}▵\left(\mathbf{2}▵+\mathbf{1}\right)\left(\mathbf{2}▵+\mathbf{3}\right)+\mathbf{1}\right]+\mathbf{1}$	Decomposition of  $\left({\mathit{GM}}_{\mathit{a},\mathbf{11}}-\mathbf{1}\right)$
2047	$=2\ast 11\ast 1\ast 3\left[2\ast 1\ast 3\ast 5+1\right]+1$	$=23\ast 89=\left(2\ast 11+1\right)\left(2^3\ast 11+1\right)$
175099	$=2\ast 11\ast 3\ast 7\left[2\ast 3\ast 7\ast 9+1\right]+1$	$={23}^2\ast 331={\left(2\ast 11+1\right)}^2\left(2\ast 11\ast 3\ast 5+1\right)$
4017157	$=2\ast 11\ast 6\ast 13\left[2\ast 6\ast 13\ast 15+1\right]+1$	$=23\ast 174659=\left(2\ast 11+1\right)\left(2\ast 11\ast 17\ast 467+1\right)$
44633821	$=2\ast 11\ast 10\ast 21\left[2\ast 10\ast 21\ast 23+1\right]+1$	$=6359\ast 7019=(2\ast 11\ast {17}^2+1)(2\ast {11}^2\ast 29+1)$
313968931	$=2\ast 11\ast 15\ast 31\left[2\ast 15\ast 31\ast 33+1\right]+1$	prime
1614529687	$=2\ast 11\ast 21\ast 43\left[2\ast 21\ast 43\ast 45+1\right]+1$	$=89\ast 18140783=(2^3\ast 11+1)(2\ast 11\ast 19\ast 43399+1)$
6612607849	$=2\ast 11\ast 28\ast 57\left[2\ast 28\ast 57\ast 59+1\right]+1$	prime
22791125017	$=2\ast 11\ast 36\ast 73\left[2\ast 36\ast 73\ast 75+1\right]+1$	$=23\ast 990918479=(2\ast 11+1)(2\ast 11\ast 45041749+1)$
68618940391	$=2\ast 11\ast 45\ast 91\left[2\ast 45\ast 91\ast 93+1\right]+1$	prime

2.3. Congruence Properties of Generalized Mersenne Numbers

2.3.1. Corollary on Congruence of Generalized Mersenne Numbers

We start first with a corollary of Theorem 2.

Corollary 3.

For all natural integer bases $a\ge 2$, all Generalized Mersenne numbers are such that

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 1\left(\mathit{mod}n\right)\hfill \end{array}$$

(2.43)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 1\left(\mathit{mod}a\right)\hfill \end{array}$$

(2.44)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 1\left(\mathit{mod}\left(a-1\right)\right)\hfill \end{array}$$

(2.45)

for all natural integer prime exponents $n\ge 3$ and

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 1\left(\mathit{mod}\left(a\left(a-1\right)+1\right)\right)\hfill \end{array}$$

(2.46)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 1\left(\mathit{mod}\left(a\left(a-1\right)\left(a^2-a+1\right)\right)\right)\hfill \end{array}$$

(2.47)

for all natural integer prime exponents $n\ge 5$.

Proof. 

Let a and n be natural integers with $a\ge 2$ and n prime, $n\ge 3$. Relation (2.43) holds obviously from (2.9) or from (2.2) by Fermat’s little theorem. More precisely, all $G{M}_{a,n}$ are such that

$$G{M}_{a,n}\equiv 1\left(\mathrm{mod}2n\right)$$

(2.48)

Relations (2.44) and (2.45) are also deduced directly from (2.9) and (2.46) and (2.47) are deduced from (2.10). □

Note that for $n=2$, $G{M}_{a,2}\equiv \pm 1\left(\mathrm{mod}4\right)$ obviously as $G{M}_{a,2}$ are all odd natural integers.

2.3.2. Generalization of a First Theorem on Congruence of Mersenne Numbers

Several theorems are known on the congruence of Mersenne numbers and their factors (see e.g. [23] and [27]). These can easily be extended to Generalized Mersenne numbers.

With notations of this paper, a first theorem on Mersenne numbers states that if n is odd, $n\ge 3$, then $M_n\equiv 7\left(\mathrm{mod}12\right)$. This theorem is generalized as follows

Theorem 4.

For all natural integer bases $a\ge 2$, and for all natural integer prime exponents $n\ge 3$, all Generalized Mersenne numbers are such that

$$G{M}_{a,n}\equiv 1\left(\mathit{mod}6\right)$$

(2.49)

and more precisely,

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 1\left(\mathit{mod}12\right)ifa\equiv 0\left(\mathit{mod}4\right)or1\left(\mathit{mod}4\right)\hfill \end{array}$$

(2.50)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 7\left(\mathit{mod}12\right)ifa\equiv 2\left(\mathit{mod}4\right)or3\left(\mathit{mod}4\right)\hfill \end{array}$$

(2.51)

Proof. 

Let a, n, m be natural integers with $a\ge 2$ and n prime, $n\ge 3$. Noting first from (2.9) and (2.48) that one can write in all generality

$$G{M}_{a,n}\equiv \left(\left(2n-m\right)Q_n\left(a\right)+1\right)\left(\mathrm{mod}m\right)\equiv \left(2n{Q}_n\left(a\right)+1\right)\left(\mathrm{mod}m\right)$$

(2.52)

(2.49) is easily demonstrated for $m=6$. For $n=3$, (2.52) reduces directly to (2.49). For $n\ge 5$, replacing $Q_n\left(a\right)$ in (2.52) by (2.10) yields

$$G{M}_{a,n}\equiv \left(2n{Q}_n\left(a\right)+1\right)\left(\mathrm{mod}6\right)\equiv \left(2n\left(▵\left(2▵+1\right)Q{"}_n\left(2▵\right)\right)+1\right)\left(\mathrm{mod}6\right)$$

(2.53)

As $▵\left(a-1\right)$ is a multiple of 3 if $a\equiv 0\left(\mathrm{mod}3\right)$ or $1\left(\mathrm{mod}3\right)$ and $\left(2▵\left(a-1\right)+1\right)$ is a multiple of 3 if $a\equiv 2\left(\mathrm{mod}3\right)$, (2.49) holds for all values of bases a and for all odd prime exponents n.

Relations (2.50) and (2.51) are demonstrated similarly for $m=12$ in (2.52). □

2.3.3. Theorem on congruence of Generalized Mersenne numbers

A new theorem on Generalized Mersenne numbers is proposed as follows.

Theorem 5.

For all natural integer bases $a\ge 2$, and for natural integer prime exponents $n\ge 3$, all Generalized Mersenne numbers are such that, if $n\equiv 1\left(\mathrm{mod}4\right)$,

$$G{M}_{a,n}\equiv 1\left(\mathit{mod}10\right)$$

(2.54)

and, if $n\equiv 3\left(\mathit{mod}4\right)$,

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 1\left(\mathit{mod}10\right)ifa\equiv 0\left(\mathit{mod}5\right)or1\left(\mathit{mod}5\right)\hfill \end{array}$$

(2.55)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 7\left(mod10\right)ifa\equiv 2\left(\mathit{mod}5\right)or4\left(\mathit{mod}5\right)\hfill \end{array}$$

(2.56)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 9\left(mod10\right)ifa\equiv 3\left(\mathit{mod}5\right)\hfill \end{array}$$

(2.57)

Proof. 

Let a, i, k, n, m, r be natural integers with $a\ge 2$ and n prime, $n\ge 3$, and let $\alpha$ be an integer and ▵ is written for $▵\left(a-1\right)$ for convenience.

(i) For the first case $n=4r+1$, (2.54) is immediate for $n=5$ by (2.9).

For $n>5$, we have to show that $Q_n\left(a\right)\equiv 0\left(\mathrm{mod}5\right)$ or $Q_n^{\prime }\left(2▵\right)\equiv 0\left(\mathrm{mod}5\right)$. If $a\equiv 0\left(\mathrm{mod}5\right)$ or $1\left(\mathrm{mod}5\right)$, then $▵\left(a-1\right)\equiv 0\left(\mathrm{mod}5\right)$ and (2.54) is proven by (2.10) for these two cases.

For $a\equiv 3\left(\mathrm{mod}5\right)$, one has $2▵\left(a-1\right)\equiv 1\left(\mathrm{mod}5\right)$ and (2.28) yields

$$Q_n^{\prime }\left(2▵\right)\equiv \left(\sum _{k=1}^{\frac{n-1}{2}}\left[\frac{C_{k-1}^{n-k-1}}{k}\right]\right)\left(\mathrm{mod}5\right)$$

(2.58)

Hensley established [14] a computer assisted proof, based on the Zeilberger algorithm, and confirmed by a Maple computation, showing that

$$\begin{array}{cc}\hfill \sum _{k=1}^m\left[{\alpha }^{k-1}\frac{C_{k-1}^{2m-k}}{k}\right]=& \frac{-1}{\alpha \left(2m+1\right)}\left(1+2^{-\left(m+1\right)}\left(\frac{{\left(1+2\alpha -\sqrt{1+4\alpha }\right)}^m\left(1+4\alpha -\sqrt{1+4\alpha }\right)}{\sqrt{1+4\alpha }}\right.\right.\hfill \\ \hfill & \left.\left.-\frac{{\left(1+2\alpha +\sqrt{1+4\alpha }\right)}^m\left(1+4\alpha +\sqrt{1+4\alpha }\right)}{\sqrt{1+4\alpha }}\right)\right)\hfill \end{array}$$

(2.59)

For $m=\frac{n-1}{2}$ and $\alpha =1$, (2.59) simplifies into

$$\begin{array}{cc}\hfill \sum _{k=1}^{\frac{n-1}{2}}\left[\frac{C_{k-1}^{n-k-1}}{k}\right]& =\frac{-1}{n}\left(1+2^{\frac{n+1}{2}}\left(\frac{{\left(3-\sqrt{5}\right)}^{\frac{n-1}{2}}\left(5-\sqrt{5}\right)}{\sqrt{5}}-\frac{{\left(3+\sqrt{5}\right)}^{\frac{n-1}{2}}\left(5+\sqrt{5}\right)}{\sqrt{5}}\right)\right)\hfill \\ \hfill & =\frac{\left({\left(\frac{1+\sqrt{5}}{2}\right)}^n+{\left(\frac{1-\sqrt{5}}{2}\right)}^n\right)-1}{n}=\frac{L_n-1}{n}\hfill \end{array}$$

(2.60)

where the definition of Lucas numbers $L_n$ in function of the golden ratio was used (see relations (62) in [33] and (70) in [12]). This expression (2.60) is always $0\left(\mathrm{mod}5\right)$ for n prime of the form $n=4r+1$, as

$$\frac{L_n-1}{n}=\frac{L_{4r+1}-L_1}{4r+1}=\frac{5F_{2r}{F}_{2r+1}}{4r+1}=\frac{5{\sum }_{i=1}^{2r}\left[F_i^2\right]}{4r+1}$$

(2.61)

where transformations of Lucas numbers $L_n$ into Fibonacci numbers $F_i$ were used (see relations (17b) and (45) in [33], (5) and (23) in [12], (13) and (14) in [2], (77) in [15], identity 9 in [20], and [21]). Furthermore, it was shown [22,26] that the product of two consecutive Fibonacci numbers $F_{2r}{F}_{2r+1}$ for $n=4r+1$ prime $>5$ is divisible by n. Therefore, $Q_n^{\prime }\left(2▵\right)\equiv 0\left(\mathrm{mod}5\right)$ in (2.58) and (2.54) holds for $a\equiv 3\left(\mathrm{mod}5\right)$.

For $a\equiv 2\left(\mathrm{mod}5\right)$ and $a\equiv 4\left(\mathrm{mod}5\right)$, one has $2▵\left(a-1\right)\equiv 2\left(\mathrm{mod}5\right)$ and (2.28) yields

$$Q_n^{\prime }\left(2▵\right)\equiv \left(\sum _{k=1}^{\frac{n-1}{2}}\left[2^{k-1}\frac{C_{k-1}^{n-k-1}}{k}\right]\right)\left(\mathrm{mod}5\right)$$

(2.62)

For $m=\frac{n-1}{2}$ and $\alpha =2$, (2.59) simplifies directly into

$$\sum _{k=1}^{\frac{n-1}{2}}\left[2^{k-1}\frac{C_{k-1}^{n-k-1}}{k}\right]=\frac{2^{n-1}-1}{n}$$

(2.63)

which is always congruent to 0 modulo 5 for n prime of the form $n=4r+1$. Indeed, one has $\left(2^{4r}\text{–}1\right)=\left({16}^r\text{–}1\right)$, and as ${16}^r\equiv 6\left(\mathrm{mod}10\right)$, then $\left(2^{4r}\text{–}1\right)\equiv 5\left(\mathrm{mod}10\right)$, meaning that $(2^{n-1}-1)$ is a multiple of 5. Furthermore, by Fermat’s theorem, $(2^{n-1}-1)$ is always divisible by n if n is prime. Therefore $\left(\frac{2^{n-1}-1}{n}\right)\equiv 0\left(\mathrm{mod}5\right)$ and relation (2.62) holds for n prime of the form $n=4r+1$. The theorem first part (2.54) is then proven for all cases of a.

(ii) For $n=4r+3$, if $a\equiv 0\left(\mathrm{mod}5\right)$ or $1\left(\mathrm{mod}5\right)$, then $▵\left(a-1\right)\equiv 0\left(\mathrm{mod}5\right)$ and the theorem second part (2.55) is proven by (2.10) for these two cases like above.

For $a\equiv 3\left(\mathrm{mod}5\right)$, one has $2▵\left(a-1\right)\equiv 1\left(\mathrm{mod}5\right)$ and we need to show (2.57). From (2.9), it is sufficient to show that the product $n▵Q_n^{\prime }\left(2▵\right)\equiv 4\left(mod5\right)$. Relation (2.28) yields

$$n▵Q_n^{\prime }\left(2▵\right)\equiv \left(3n\sum _{k=1}^{\frac{n-1}{2}}\left[\frac{C_{k-1}^{n-k-1}}{k}\right]\right)\left(\mathrm{mod}5\right)$$

(2.64)

With (2.60), it yields

$$3n\sum _{k=1}^{\frac{n-1}{2}}\left[\frac{C_{k-1}^{n-k-1}}{k}\right]=3\left(L_n-1\right)$$

(2.65)

which is always congruent to 4 modulo 5 for all values of n of the form $n=4r+3$, as one has

$$L_{4r+3}\text{–}1=5F_{2r+1}{F}_{2r+2}-2$$

(2.66)

as a special case of relations (17a) and (17b) in [33], meaning that (2.65) is always a multiple of 15 minus 6, i.e. always congruent to 4 modulo 5.

For $a\equiv 2\left(\mathrm{mod}5\right)$ and $a\equiv 4\left(\mathrm{mod}5\right)$, one has $2▵\left(a-1\right)\equiv 2\left(\mathrm{mod}5\right)$ and we need to show (2.56). Again, from (2.9), it is sufficient to show that the product $n▵Q_n^{\prime }\left(2\Delta \right)\equiv 3\left(\mathrm{mod}5\right)$. Relation (2.28) yields then

$$n▵Q_n^{\prime }\left(2▵\right)\equiv \left(n\sum _{k=1}^{\frac{n-1}{2}}\left[2^{k-1}\frac{C_{k-1}^{n-k-1}}{k}\right]\right)\left(\mathrm{mod}5\right)$$

(2.67)

From (76), one has

$$n\sum _{k=1}^{\frac{n-1}{2}}\left[2^{k-1}\frac{C_{k-1}^{n-k-1}}{k}\right]=2^{n-1}-1$$

(2.68)

which is always congruent to 3 modulo 5 for all values of n of the form $n=4r+3$. As above, one has then $\left(2^{4r+2}\text{–}1\right)=\left(4\times {16}^r-1\right)\equiv 3\left(\mathrm{mod}10\right)$, as $(4\times {16}^r)\equiv 4\left(\mathrm{mod}10\right)$. Therefore (2.67) holds for any values of n of the form $n=4r+3$. The theorem second part (2.55) to (2.57) is proven for all cases of bases a. □

2.4. Congruence Properties of Generalized Mersenne Numbers and Their Factors

2.4.1. Generalization of a Second Theorem on Mersenne Numbers

For Generalized Mersenne composites, let’s note generally their positive natural integer factors $c_i$ such as

$$G{M}_{a,n}=c_1^{e_1}{c}_2^{e_2}...c_i^{e_i}...$$

(2.69)

where $e_i$ are positive natural integer exponents. A theorem on factors of Mersenne numbers states, with the notations in this paper, that if n is an odd prime and if $c_i$ divides $M_n$, then $c_i\equiv 1\left(\mathrm{mod}n\right)$ and $c_i\equiv \pm 1\left(\mathrm{mod}8\right)$.

The first part is not only obviously true for all $M_n$ by (2.3), but can be generalized to $c_i\equiv 1\left(\mathrm{mod}2n\right)$. The second part is also obviously correct for factors $c_i$ of Mersenne numbers $M_n$ noting that, first, all $M_n\equiv -1\left(mod8\right)$ for $n\ge 3$; second, at least one of the factors $c_i$ of the Mersenne number $M_n=G{M}_{2,n}$ must be congruent to $-1$ modulo 8; and third that the sum of exponents $e_i$ of factors $c_i$ which are congruent to $-1$ modulo 8 must be odd. This is however no longer correct for all $G{M}_{a,n}$ with $a>2$.

This theorem can be generalized in two steps. The first part is generalized in the following Theorem.

Theorem 6.

For all natural integer bases $a\ge 2$, if n is an odd prime and if a positive natural integer $c_i$ divides $G{M}_{a,n}$, then

$$c_i\equiv 1\left(mod2n\right)$$

(2.70)

Proof. 

Let a, b, n, m, i, k, $c_i$, $f_i$, $f_i^{\prime }$, ${\lambda }_i$, $r_i$, p, q be natural integers with $a\ge 2$, n prime, $n\ge 3$, $m>1$, $k>0$, $c_i\ge 1$, p prime, $q>0$ and $1\le i\le q$.

Proving this theorem is equivalent to show that all prime integer factors of $G{M}_{a,n}$ are of the form

$$c_i=2n{f}_i+1$$

(2.71)

Let us assume first the contrary, i.e. that the prime integer factors $c_i$ of $G{M}_{a,n}$ are not of the form (2.71). For q factors $c_i$ (the case where their exponents $e_i\ne 1$ can be treated similarly), one has from (2.9) and (2.48)

$$G{M}_{a,n}=c_1{c}_2\dots c_q=2n{Q}_n\left(a\right)+1\equiv 1\left(\mathrm{mod}2n\right)$$

(2.72)

Let us then write generally

$$c_i=2n{f}_i^{\prime }+{\lambda }_i$$

(2.73)

with the condition that the product

$${\lambda }_1{\lambda }_2\dots {\lambda }_q\equiv 1\left(\mathrm{mod}2n\right)$$

(2.74)

i.e. that all ${\lambda }_i$ are such that ${\lambda }_i\equiv 1\left(\mathrm{mod}2n\right)$ or that an even number of ${\lambda }_i$ are such that ${\lambda }_i\equiv -1\left(\mathrm{mod}2n\right)$, which means that there exist natural integers $r_i$ such as ${\lambda }_i=2n{r}_i+1$ or ${\lambda }_i=2n{r}_i-1$. Then one can write the factors $c_i$ as

$$c_i=2n\left(f_i^{\prime }+r_i\right)+1or{c}_i=2n\left(f_i^{\prime }+r_i\right)-1$$

(2.75)

Let us now assume that an even number of prime factors are of the form $c_i=2n{f}_i-1$. But this is not possible, as it was proven (see [27], p. 267, Nr 2) that all prime factors of $\left(a^m-b^m\right)$, with $a>b$ and $m>1$, are of the form $\left(mk+1\right)$. This is simply shown considering that if a prime p divides $\left(a^m-b^m\right)$, and if p does not divide a and b, then by Fermat’s theorem, p divides $\left(a^{p-1}-1\right)$ and $\left(b^{p-1}-1\right)$ and then also $\left(a^{p-1}-b^{p-1}\right)$ and therefore m divides $\left(p-1\right)$, i.e. $p=mk+1$.

For $b=\left(a-1\right)$, $m=n$ prime and $k=2f_i$, it is seen directly that n divides $\left(c_i-1\right)$ if $c_i$ is of the form (2.71). Therefore all prime integer factors of $G{M}_{a,n}$ are of the form (2.71). Furthermore, composite factors of $G{M}_{a,n}$ are also obviously of the form (2.71), being the product of prime factors of the form (2.71). □

Note that for $n=2$, all factors $c_i$ of $G{M}_{a,2}$ are obviously such that $c_i\equiv \pm 1\left(\mathrm{mod}4\right)$.

The second part of the generalization of the theorem on factors of Mersenne numbers needs to specify the congruence of $G{M}_{a,n}$ modulo 8 as in the following Theorem.

Theorem 7.

For all natural integer bases $a\ge 2$ and all prime integer exponents $n\ge 3$, all $G{M}_{a,n}$ are such that

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 1\left(\mathit{mod}8\right)ifa\equiv 0\left(\mathit{mod}8\right)or1\left(\mathit{mod}8\right)\hfill \end{array}$$

(2.76)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv -1\left(\mathit{mod}8\right)ifa\equiv -1\left(\mathit{mod}8\right)or2\left(\mathit{mod}8\right)\hfill \end{array}$$

(2.77)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv 3\left(\mathit{mod}8\right)ifa\equiv -2\left(\mathit{mod}8\right)or3\left(\mathit{mod}8\right)\hfill \end{array}$$

(2.78)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv -3\left(\mathit{mod}8\right)ifa\equiv -3\left(\mathit{mod}8\right)or4\left(\mathit{mod}8\right)\hfill \end{array}$$

(2.79)

and the factors $c_i$ of $G{M}_{a,n}$ are such that $c_i\equiv \pm 1\left(\mathit{mod}8\right)$ or $\pm 3\left(\mathit{mod}8\right)$ such that their product satisfy above relations.

Proof. 

Let a, n, m, r be natural integers with $a\ge 2$, n prime, $n\ge 3$.

The first part of this theorem is easily demonstrated, noting first from (2.27) that one has

$$Q_n\left(a\right)=▵\left(S_{n-2}^{\left(1\right)}+2▵S_{n-4}^{\left(2\right)}+4{▵}^2{S}_{n-6}^{\left(3\right)}+...\right)$$

(2.80)

Searching for the congruence conditions for $m=8$ in (2.52), it yields

$$G{M}_{a,n}\equiv \left(1+n\left(2▵S_{n-2}^{\left(1\right)}+4{▵}^2{S}_{n-4}^{\left(2\right)}+8{▵}^3{S}_{n-6}^{\left(3\right)}+...\right)\right)\left(\mathrm{mod}8\right)$$

(2.81)

As the coefficient of ${▵}^3$ and the coefficients of all terms in ▵ of higher degrees are multiple of 8, these terms can be removed from (2.81), and replacing by (2.12) and (2.14), it follows that

$$G{M}_{a,n}\equiv \left(1+2n▵+2n\left(n-3\right){▵}^2\right)\left(\mathrm{mod}8\right)$$

(2.82)

The exponent n being an odd prime, it must be either $n=4r+1$ or $n=4r+3$, yielding respectively from (2.82)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv \left(1+2▵-4{▵}^2\right)\left(\mathrm{mod}8\right)forn=4r+1\hfill \end{array}$$

(2.83)

$$\begin{array}{cc}\hfill G{M}_{a,n}& \equiv \left(1+6▵\right)\left(\mathrm{mod}8\right)forn=4r+3\hfill \end{array}$$

(2.84)

Calculating the values of $\left(1+2▵-4{▵}^2\right)\left(\mathrm{mod}8\right)$ and of $\left(1+6▵\right)\left(\mathrm{mod}8\right)$ for the first eight values of a, $2\le a\le 9$, yields the series $-1$, $+3$, $-3$, $-3$, $+3$, $-1$, $+1$, $+1$, that repeats itself indefinitely for successive values of $a>9$, hence relations (2.76) to (2.79) hold.

The second part of the theorem on the congruence of factors $c_i$ of $G{M}_{a,n}$ is then obvious. □

The factorization of the first composites $G{M}_{a,n}$ is indicated in Tables 2 for n primes, $3\le n\le 11$. It is seen that all the factors $c_i$ of composites $G{M}_{a,n}$ are of the form (2.71) and are either $G{M}_{a,n}\equiv \pm 1\left(mod8\right)$ or $\pm 3\left(mod8\right)$ such as their products satisfy relations (2.76) to (2.79).

Composite $G{M}_{a,n}$ can be written generally in function of their prime integer factors, from (2.69) and (2.71),

$$G{M}_{a,n}=c_1^{e_1}{c}_2^{e_2}...c_i^{e_i}...={\left(2n{f}_1+1\right)}^{e_1}{\left(2n{f}_2+1\right)}^{e_2}...{\left(2n{f}_i+1\right)}^{e_i}...$$

(2.85)

In the case of more than two prime integer factors and for exponents $e_i\ne 1$, a composite $G{M}_{a,n}$ can also be written in all generality as the product of two factors not necessarily primes and with their exponents $e_i=1$, as any combination of products of factors $c_i$ of the form (2.71) will be of the same form (2.71)

$$G{M}_{a,n}=c_1{c}_2=\left(2n{f}_1+1\right)\left(2n{f}_2+1\right)$$

(2.86)

Therefore, a corollary of the above Theorem 7 is as follows.

Corollary 8.

For all natural integer bases $a\ge 2$ and all prime integer exponents $n\ge 3$, a natural integer $c_i=\left(2n{f}_i+1\right)$ divides a $G{M}_{a,n}$ if and only if the integer function $Q_n\left(a\right)$ associated to the $G{M}_{a,n}$ is such that

$$Q_n\left(a\right)\equiv f_i\left(\mathit{mod}{c}_i\right)$$

(2.87)

for all factors $c_i$ and where $f_i$ are natural integers.

Proof. 

Let a, n, r be natural integers with $a\ge 2$, n prime, $n\ge 3$.

Relation (2.87) obviously holds whether $G{M}_{a,n}$ is prime or composite. For two factors like in (2.86), one has

$$G{M}_{a,n}=2n{Q}_n\left(a\right)+1=\left(2n{f}_1+1\right)\left(2n{f}_2+1\right)=2n\left(f_2{c}_1+f_1\right)+1$$

yielding immediately (2.87). If $G{M}_{a,n}$ is prime, then $f_2=0$ and $f_1=Q_n\left(a\right).$

Conversely, if the integer function $Q_n\left(a\right)$ is such that (2.87) holds with $c_1=(2n{f}_1+1)$, then it exists an integer r such as

$$Q_n\left(a\right)=r{c}_1+f_1$$

(2.88)

yielding

$$2n{Q}_n\left(a\right)+1=2nr{c}_1+2n{f}_1+1=(2n{f}_1+1)(2nr+1)=c_1{c}_2=G{M}_{a,n}$$

(2.89)

meaning that $c_1$ divides $G{M}_{a,n}$ for an appropriate choice of the integer r, which is here $f_2$ in the second factor $c_2$ of $G{M}_{a,n}$. This relation (2.87) is true whether the factors $c_1$ and $c_2$ are composites or primes of the form (2.71). □

From Table 2, it is seen that the integers $f_1$, $f_2$, ..., $f_i$, ... in (2.85) for a particular prime exponent n are increasing from one composite number to the next for increasing values of the base a and can be found in function of the integer functions $Q_n\left(a\right)$.

2.4.2. Generalization of a Third Theorem on Mersenne Numbers (Euler Theorem)

Another theorem on Mersenne numbers was stated by Euler in 1750. With the notations in this paper, it reads: if n is prime, $n\equiv 3\left(\mathrm{mod}4\right)$, then $\left(2n+1\right)$ divides $M_n$ if and only if $\left(2n+1\right)$ is a prime; in this case, if $n>3$, then $M_n$ is composite. This means that for $n\equiv 3\left(\mathrm{mod}4\right)$ and prime, $M_n=G{M}_{2,n}$ has the factor $c_1=(2n{f}_1+1)$ with $f_1=1$, and that $c_1$ in this case is prime. This is exactly the case for $n=3$ and $M_3=G{M}_{2,3}=7$; $n=11$ and $M_{11}=G{M}_{2,11}=2047=23\times 89$; and so on. This can be generalized for all $G{M}_{a,n}$ for odd primes n, irrespective of n being congruent to $3\left(mod4\right)$ or not, in the following theorem.

Theorem 9.

For all natural integer bases $a\ge 2$ and all prime integer exponents $n\ge 3$, a natural integer $c_i=\left(2n{f}_i+1\right)$ divides $G{M}_{a,n}$ if and only if, for some natural integer values of $f_i$, $c_i=\left(2n{f}_i+1\right)$ is prime or a composite formed by the product of primes of the form $\left(2nj+1\right)$, with i and j natural integers.

Before demonstrating this theorem, it is important to realize that not all integer values of $f_i$ will do in Theorem 9, only those that render the factor $c_i$ prime or composite of the form $\left(2n{f}_i+1\right)$ will be acceptable. All other integer values of $f_i$ are excluded and are called excluded values. The following Lemma is demonstrated giving the form that factors $c_i$ cannot take and the form of excluded values of $f_i$.

Lemma 10.

For all natural integer bases $a\ge 2$ and all prime integer exponents $n\ge 3$, a natural integer $c_i=\left(2n{f}_i+1\right)$ divides a $G{M}_{a,n}$ if $c_i$ and $f_i$ are different from excluded values, i.e. different respectively from either (i)

$$c_i\not\equiv 0\left(\mathit{mod}\left(2nk+1\right)\right)\mathrm{and}{f}_i\not\equiv k\left(\mathrm{mod}\left(2nk+1\right)\right)$$

(2.90)

for positive natural integers $k=2nuv+u\epsilon +v\delta +r$, with u, v and r positive natural integers such as $uv\ne 0$, ε and δ integers $\ne 0$ and $\ne 1$ and such as $\epsilon \delta \equiv 1\left(\mathit{mod}2n\right)=2nr+1$; or (ii)

$$c_i\not\equiv 0\left(\mathit{mod}\left(2nk-1\right)\right)\mathrm{and}{f}_i\not\equiv -k\left(\mathit{mod}\left(2nk-1\right)\right)$$

(2.91)

for positive natural integers k; or (iii)

$$c_i\not\equiv 0\left(\mathit{mod}\left(2nk\pm t\right)\right)\mathrm{and}{f}_i\not\equiv \left(\alpha +k\beta \right)\left(\mathit{mod}\left(2nk+\gamma \right)\right)$$

(2.92)

for natural integers k, for odd natural integers t such that $1<t<n$, for integers α, β, γ, with β and γ odd integers and $2n\alpha +1=\beta \gamma$.

Proof. 

Let a, n, i, j, k, $c_i$, $f_i$, s, u, v, x, y be natural integers with $a\ge 2$, n prime, $n\ge 3$, and $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, r integers and $\delta \ne 0$ and $\epsilon \ne 0$.

From Theorem 6, factors $c_i$ of a $G{M}_{a,n}$ are

$$c_i=2n{f}_i+1\equiv 1\left(\mathrm{mod}2n\right)$$

(2.93)

Let’s assume in all generality that $f_i$ can be written as

$$f_i\equiv x\left(\mathrm{mod}y\right)$$

(2.94)

for yet unknown natural integers x and y. For a given prime n, for $f_i$ to be excluded values, (2.93) must not be verified for all bases a. Among all possible values of $f_i$, it will be the case if in (2.93)

$$c_i=2n{f}_i+1\equiv 0\left(\mathrm{mod}y\right)$$

(2.95)

meaning that

$$2nx+1\equiv 0\left(\mathrm{mod}y\right)$$

(2.96)

is a multiple of y. Writing in all generality $x=\left(\alpha +k\beta \right)$ and $y=\left(2nk+\gamma \right)$, one has from (2.94)

$$f_i\equiv \left(\alpha +k\beta \right)\left(\mathrm{mod}\left(2nk+\gamma \right)\right)=\left(2nk+\gamma \right)s+\left(\alpha +k\beta \right)$$

(2.97)

with $\alpha$, $\beta$, $\gamma$ integers and k and s natural integers. Replacing in (2.96) yields

$$2n\left(\alpha +k\beta \right)+1\equiv 0\left(\mathrm{mod}\left(2nk+\gamma \right)\right)$$

(2.98)

or

$$\beta \left(\left(\frac{2n\alpha +1}{\beta }\right)+2nk\right)\equiv 0\left(\mathrm{mod}\left(2nk+\gamma \right)\right)$$

(2.99)

which gives the condition

$$2n\alpha +1=\beta \gamma$$

(2.100)

where $\beta$ and $\gamma$ are obviously odd integers, either positive and/or negative depending on the sign of $\alpha$. The factors $c_i$ read then from (2.95) and (2.97) with (2.100)

$$c_i=2n\left(\left(2nk+\gamma \right)s+\left(\alpha +k\beta \right)\right)+1=\left(2nk+\gamma \right)\left(2ns+\beta \right)$$

(2.101)

All $f_i$ of the form (2.97) are excluded values and all $c_i$ of the form (2.101) cannot be factors of $G{M}_{a,n}$ for every integers $\alpha$, $\beta$, $\gamma$ complying with (2.100) and for all natural integers k, except for the following specific cases.

(i) First, for the triplet $\left(\alpha ,\beta ,\gamma \right)=\left(0,1,1\right)$ verifying (2.100), $f_i$ (2.97) and factors $c_i$ (2.101) read respectively

$$f_i\equiv k\left(\mathrm{mod}\left(2nk+1\right)\right)=\left(2nk+1\right)s+k$$

(2.102)

$$c_i=2n\left(\left(2nk+1\right)s+k\right)+1=\left(2nk+1\right)\left(2ns+1\right)$$

(2.103)

If for certain positive integers k, $\left(2nk+1\right)$ is prime, then by Theorem 6, $c_i$ (2.103) are factors of a $G{M}_{a,n}$ and $f_i$ (2.102) are not excluded values.

If for other positive integers k, $\left(2nk+1\right)$ is composite, it can be written as

$$\left(2nk+1\right)=\left(2nu+\delta \right)\left(2nv+\epsilon \right)$$

(2.104)

with the obvious condition

$$\delta \epsilon \equiv 1\left(mod2n\right)=2nr+1$$

(2.105)

where u and v are natural integers with u and v not simultaneously null; $\delta$, $\epsilon$ and r are integers with $\delta \ne 0$ and $\epsilon \ne 0$; and

$$k=2nuv+u\epsilon +v\delta +r$$

(2.106)

As k must be a natural integer, only the values of $\delta$ and $\epsilon$ complying with (2.105) must be considered. For $\delta =\epsilon =1$ (i.e., $r=0$), $k=2nuv+u+v$ and the factors of $\left(2nk+1\right)$ are

$$\left(2nk+1\right)=\left(2nu+1\right)\left(2nv+1\right)$$

(2.107)

showing that $f_i$ (2.102) with (2.107) are not excluded values, similarly to the above case of $\left(2nk+1\right)$ being prime.

For all the other cases of values of k in (2.106) with $\delta$ and $\epsilon$ integers $\ne 0$ and $\ne 1$, and complying with (2.105), the factors $c_i$ from (2.103) read

$$c_i=\left(2ns+1\right)\left(2nu+\delta \right)\left(2nv+ϵ\right)$$

(2.108)

which, by Theorem 6, cannot be factors of a $G{M}_{a,n}$ and the corresponding $f_i$ (2.102) are excluded values. For example, with $\delta =\epsilon =-1$ (i.e. $r=0$), the factors of $\left(2nk+1\right)$ are $\left(2nu-1\right)\left(2nv-1\right)$, showing from (2.103) that $c_i=\left(2ns+1\right)\left(2nu-1\right)\left(2nv-1\right)$ cannot be factors of a $G{M}_{a,n}$ and that the corresponding $f_i$ are excluded values.

(ii) Second, for the triplet $\left(\alpha ,\beta ,\gamma \right)=\left(0,-1,-1\right)$ verifying (2.100), $f_i$ (2.97) and factors $c_i$ (2.101) read respectively

$$f_i\equiv -k\left(\mathrm{mod}\left(2nk-1\right)\right)=\left(2nk-1\right)s-k$$

(2.109)

$$c_i=2n\left(\left(2nk-1\right)s-k\right)+1=\left(2nk-1\right)\left(2ns-1\right)$$

(2.110)

showing again by Theorem 6 that $c_i$ (2.110) cannot be factors of a $G{M}_{a,n}$ and that $f_i$ (2.109) are excluded values for all positive integers k.

(iii) Third, for the general case where $\alpha \ne 0$, from (2.100), both $\beta$ and $\gamma$ are obviously $\ne 1$ and therefore again by Theorem 6, $c_i$ (2.101) cannot be factors of a $G{M}_{a,n}$ and all $f_i$ (2.97) are excluded values for all natural integers k.

Summarizing, the excluded values of $f_i$ and the excluded forms of factors $c_i$ are respectively (2.90) for positive integers k (2.106) with $\delta$ and $\epsilon$ integers $\ne 0$ and $\ne 1$; (2.91) for all positive integers k; and (2.92) for all integers $\alpha$, all odd integers $\beta$ and $\gamma$ complying with (2.100), all natural integers k and all t odd integers such that $1<t<n$, as, from the form of factors $c_i$ (2.101),

$$t\equiv \beta \left(\mathrm{mod}2nk\right)ort\equiv \gamma \left(\mathrm{mod}2nk\right)$$

(2.111)

The excluded forms of factors $c_i$ (2.92) are always composites and the product of at least two factors, which are multiple of integers of the form $\left(2nj-1\right)$ and/or $\left(2nj\pm t\right)$ with j natural integers and at least once $j=k$. □

Theorem 9 can now be demonstrated as follows.

Proof. 

Let $a,n,i,j,k,c_i,f_i$ be natural integers with $a\ge 2$, n prime, $n\ge 3$.

The first part of the demonstration is quite straightforward as from Theorem 6 above, all natural integer prime and composite factors of $G{M}_{a,n}$ are of the form (2.71).

Conversely, if an natural integer $c_1=\left(2n{f}_1+1\right)$ is prime or a composite formed by the product of primes of the form $\left(2nj+1\right)$, then, for a suitable choice of an integer $f_2$, a natural integer function ${\mathsf{\Phi }}_n$ can be found and written as

$${\mathsf{\Phi }}_n=f_2\left(2n{f}_1+1\right)+f_1$$

(2.112)

The suitable choice of the integer $f_2$ means here that it must not be an excluded value specifically for the prime exponent n as shown in Lemma 10, i.e. that $c_2=\left(2n{f}_2+1\right)$ must itself be either a prime or a composite formed by the product of primes of the form $\left(2nj+1\right)$. Relation (2.112) then yields

$${\mathsf{\Phi }}_n\equiv f_1\left(\mathrm{mod}\left(2n{f}_1+1\right)\right)$$

(2.113)

and by the Corollary 8 above, $c_1=\left(2n{f}_1+1\right)$ divides $G{M}_{a,n}=\left(2n{\mathsf{\Phi }}_n+1\right)$, i.e. there is a base a for which the integer function $Q_n\left(a\right)$ in (2.4) specific for each prime exponent n is equal to ${\mathsf{\Phi }}_n$ (2.112). □

We emphasize that not all integer values of $f_1$ and $f_2$ will do and that the integer $f_2$ must be chosen suitably, such that the factors $c_1=\left(2n{f}_1+1\right)$ and $c_2=\left(2n{f}_2+1\right)$ are prime or composite formed by the product of primes of the form $\left(2nj+1\right)$. All other values of $f_1$ and $f_2$ are excluded values as shown in Lemma 10.

2.4.3. Theorem on congruence of coefficients $f_1$ and $f_2$

The form of the integers $f_1$ and $f_2$ in the factors $c_1$ and $c_2$ of composite $G{M}_{a,n}$ can be determined in function of the exponent n, the base a and the factors $c_1$ and $c_2$ by the following theorem.

Theorem 11.

If a composite $G{M}_{a,n}$ has $c_1=\left(2n{f}_1+1\right)$ and $c_2=\left(2n{f}_2+1\right)$ as two factors, then $f_1\equiv u\left(\mathrm{mod}4\right)$ and $f_2\equiv v\left(\mathrm{mod}4\right)$ with u and $v=0$, 1, 2 or 3, depending on congruence of $n\left(\mathrm{mod}4\right)$ and on congruence of $a\left(\mathrm{mod}8\right)$, as shown in Table 3.

Table 3. Congruence of natural integers $f_1$ and $f_2$$\left(\mathrm{mod}4\right)$.

For $n\equiv 1\left(\mathrm{mod}4\right)$
$c_1\equiv$	$f_1\equiv$	if $a\equiv 0$ or $1\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 2$ or $7\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 3$ or $6\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 4$ or $5\left(\mathrm{mod}8\right)$ $f_2\equiv$
$1\left(\mathrm{mod}8\right)$	$0\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$
$3\left(\mathrm{mod}8\right)$	$1\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$
$5\left(\mathrm{mod}8\right)$	$2\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$
$7\left(\mathrm{mod}8\right)$	$3\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$
For $n\equiv 3\left(\mathrm{mod}4\right)$
$c_1\equiv$	$f_1\equiv$	if $a\equiv 0$ or $1\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 2$ or $7\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 3$ or $6\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 4$ or $5\left(\mathrm{mod}8\right)$ $f_2\equiv$
$1\left(\mathrm{mod}8\right)$	$0\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$
$3\left(\mathrm{mod}8\right)$	$3\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$
$5\left(\mathrm{mod}8\right)$	$2\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$
$7\left(\mathrm{mod}8\right)$	$1\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$

Table 3. Congruence of natural integers $f_1$ and $f_2$$\left(\mathrm{mod}4\right)$.

For $n\equiv 1\left(\mathrm{mod}4\right)$
$c_1\equiv$	$f_1\equiv$	if $a\equiv 0$ or $1\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 2$ or $7\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 3$ or $6\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 4$ or $5\left(\mathrm{mod}8\right)$ $f_2\equiv$
$1\left(\mathrm{mod}8\right)$	$0\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$
$3\left(\mathrm{mod}8\right)$	$1\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$
$5\left(\mathrm{mod}8\right)$	$2\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$
$7\left(\mathrm{mod}8\right)$	$3\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$
For $n\equiv 3\left(\mathrm{mod}4\right)$
$c_1\equiv$	$f_1\equiv$	if $a\equiv 0$ or $1\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 2$ or $7\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 3$ or $6\left(\mathrm{mod}8\right)$ $f_2\equiv$	if $a\equiv 4$ or $5\left(\mathrm{mod}8\right)$ $f_2\equiv$
$1\left(\mathrm{mod}8\right)$	$0\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$
$3\left(\mathrm{mod}8\right)$	$3\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$
$5\left(\mathrm{mod}8\right)$	$2\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$
$7\left(\mathrm{mod}8\right)$	$1\left(\mathrm{mod}4\right)$	$1\left(\mathrm{mod}4\right)$	$0\left(\mathrm{mod}4\right)$	$2\left(\mathrm{mod}4\right)$	$3\left(\mathrm{mod}4\right)$

The demonstration of this theorem is based on the above Theorems 7 and 9.

Proof. 

Let $a,n,i,j,c_i,f_i,u,v$ be natural integers with $a\ge 2$, n prime, $n\ge 3$ and $\alpha ,\beta ,\gamma$ integers. Let $c_1$ and $c_2$ be the two factors of $G{M}_{a,n}=c_1{c}_2$. From Theorem 9, $c_1$ and $c_2$ are primes of the form $\left(2n{f}_1+1\right)$ and/or composites of the form of a product of integers $\left(2nj+1\right)$. From Theorem 7, one has

$$G{M}_{a,n}\equiv \alpha \left(\mathrm{mod}8\right)$$

(2.114)

with $c_1\equiv \beta \left(\mathrm{mod}8\right)$ and $c_2\equiv \gamma \left(\mathrm{mod}8\right)$ where $\alpha$, $\beta$ and $\gamma$ take values either $\pm 1$ or $\pm 3$, with the obvious condition that

$$\alpha \equiv \beta \gamma \left(\mathrm{mod}8\right)$$

(2.115)

which then yields by Theorem 7

$$\begin{array}{cc}\hfill \beta & =\gamma for\alpha =+1\mathrm{i}.\mathrm{e}\mathrm{.},\mathrm{for}a\equiv 0\left(\mathrm{mod}8\right)\mathrm{or}1\left(\mathrm{mod}8\right)\hfill \end{array}$$

(2.116)

$$\begin{array}{cc}\hfill \beta & =-\gamma \mathrm{for}\alpha =-1\mathrm{i}.\mathrm{e}.,\mathrm{for}a\equiv 2\left(\mathrm{mod}8\right)\mathrm{or}7\left(\mathrm{mod}8\right)\hfill \end{array}$$

(2.117)

$$\begin{array}{cc}\hfill \beta & =-\gamma +4\mathrm{for}\alpha =+3\mathrm{i}.\mathrm{e}.,\mathrm{for}a\equiv 3\left(\mathrm{mod}8\right)\mathrm{or}6\left(\mathrm{mod}8\right)\hfill \end{array}$$

(2.118)

$$\begin{array}{cc}\hfill \beta & =\gamma -4\mathrm{for}\alpha =-3\mathrm{i}.\mathrm{e}.,\mathrm{for}a\equiv 4\left(\mathrm{mod}8\right)\mathrm{or}5\left(\mathrm{mod}8\right)\hfill \end{array}$$

(2.119)

For

$$c_1=2n{f}_1+1\equiv \beta \left(\mathrm{mod}8\right)$$

(2.120)

one has for

$$\begin{array}{cc}\hfill n& \equiv 1\left(\mathrm{mod}4\right):f_1\equiv \left(\frac{\beta -1}{2}\right)\left(\mathrm{mod}4\right)\equiv u\left(\mathrm{mod}4\right)\hfill \end{array}$$

(2.121)

$$\begin{array}{cc}\hfill n& \equiv 3\left(\mathrm{mod}4\right):f_1\equiv \left(\frac{1-\beta }{2}\right)\left(\mathrm{mod}4\right)\equiv u\left(\mathrm{mod}4\right)\hfill \end{array}$$

(2.122)

and $f_2\equiv v\left(\mathrm{mod}4\right)$ is found by replacing in (2.121) and (2.122) $\beta$ in function of $\gamma$ from (2.116) to (2.119) depending on the prime exponent n and the base a. Hence, congruences given in Table 3 hold. □

Note that for Mersenne numbers (i.e., for $a=2$ in Table 3), $c_1\equiv 1\left(\mathrm{mod}8\right)$ or $7\left(\mathrm{mod}8\right)$, yielding that $f_1$ and $f_2$ are congruent to $0\left(\mathrm{mod}4\right)$ and/or $3\left(\mathrm{mod}4\right)$ for $n\equiv 1\left(\mathrm{mod}4\right)$, and $f_1$ and $f_2$ are congruent to $0\left(\mathrm{mod}4\right)$ and/or $1\left(mod4\right)$ for $n\equiv 3\left(mod4\right)$.

3. Results and discussion

Distributions of primes and composites in Generalized Mersenne numbers are further investigated in two companion papers. However, Generalized Mersenne numbers as presented in this paper are useful to approach the problem of why most of the Mersenne numbers with prime exponents are not themselves primes. It was mentioned in the introduction that composite and prime Generalized Mersenne numbers appear apparently at random for different values of the exponent n and the base a. It is seen also that prime Generalized Mersenne numbers can be found for larger values of the base a for exponents n that yield Mersenne composites, like, e.g., for $n=11,23,29,...$ It appears that some exponents n are less “productive” than others to yield Generalized Mersenne primes. The reason for this is still unknown but it shows that Mersenne numbers that are composite for prime exponents are nothing exceptional and are simply Generalized Mersenne composites for $a=2$. Sequences of Generalized Mersenne numbers, primes, bases, and exponents can be found online at the Online Encyclopedia of Integer Sequences (OEIS) [28]; see Table 4.

The density of Mersenne primes is also very low. Let’s consider the largest known Mersenne prime $M_{82589933}=\left(2^{82589933}-1\right)$, having 24862048 digits,

Table 4. OEIS references of sequences of Generalized Mersenne numbers, primes, bases and exponents for k integers.

n	${\mathit{GM}}_{\mathit{a},\mathit{n}}$	${\mathit{GM}}_{\mathit{a},\mathit{n}}$ primes
	numbers	primes	$\mathit{a}$	#for $\mathit{a}\le {\mathbf{10}}^{\mathit{k}}$	$\#<{\mathbf{10}}^{\mathit{k}}$	${\mathbf{10}}^{\mathit{k}-\mathbf{1}}<\#<{\mathbf{10}}^{\mathit{k}}$
2	A005408	A000040	–	–	A006880	A006879
3	A003215	A002407	A002504	A221794	A113478	A221792
5	A022521	A121616	A121617	A221849	A221846	A221847
7	A022523	A121618	A121619	A221980	A221977	A221978
11	A022527	A189055	A211184	A221986	A221983	A221984
13	A022529	–	–	–	–	–
17	A022533	–	–	–	–	–
19	A022535	–	–	–	–	–
23	A022539	–	–	–	–	–

Notes: # means “Number of $G{M}_{a,n}$ primes”. For $n=2$, the first prime, 2, must be removed from the sequences indicated in the first row as $G{M}_{a,2}$ generates only all the odd integers. In some sequences, a shift of one unity must be applied.

Table 4. OEIS references of sequences of Generalized Mersenne numbers, primes, bases and exponents for k integers.

n	${\mathit{GM}}_{\mathit{a},\mathit{n}}$	${\mathit{GM}}_{\mathit{a},\mathit{n}}$ primes
	numbers	primes	$\mathit{a}$	#for $\mathit{a}\le {\mathbf{10}}^{\mathit{k}}$	$\#<{\mathbf{10}}^{\mathit{k}}$	${\mathbf{10}}^{\mathit{k}-\mathbf{1}}<\#<{\mathbf{10}}^{\mathit{k}}$
2	A005408	A000040	–	–	A006880	A006879
3	A003215	A002407	A002504	A221794	A113478	A221792
5	A022521	A121616	A121617	A221849	A221846	A221847
7	A022523	A121618	A121619	A221980	A221977	A221978
11	A022527	A189055	A211184	A221986	A221983	A221984
13	A022529	–	–	–	–	–
17	A022533	–	–	–	–	–
19	A022535	–	–	–	–	–
23	A022539	–	–	–	–	–

Notes: # means “Number of $G{M}_{a,n}$ primes”. For $n=2$, the first prime, 2, must be removed from the sequences indicated in the first row as $G{M}_{a,2}$ generates only all the odd integers. In some sequences, a shift of one unity must be applied.

If we compare the number of known Mersenne primes, 51, to first, the number of all the primes less than ${10}^{24862048}$ that can be approximated from the prime number theorem as $\Pi \left({10}^{24862048}\right)\approx {10}^{24862048}/ln\left({10}^{24862048}\right)$, i.e. approximately $1.75\ast {10}^{24862042}$, and second to the number of Mersenne numbers with prime exponents, i.e. the number of primes less than 82589933, i.e., $\Pi \left(82589933\right)\approx 82589933/ln\left(82589933\right)$, or approximately 4530590, we see that the density of Mersenne primes is extremely low, in the order of $2.1\ast {10}^{-24862041}$ and $1.1\ast {10}^{-5}$ respectively for the first and second cases.

Mersenne primes are used in cryptography (see, e.g., [1,3,7,18,31,32]). But to fix the ideas, only medium-sized Mersenne primes are used in cryptography. So the search for larger Mersenne primes doesn’t have applications in cryptography. Generally speaking, there are two applications of Mersenne primes within cryptography [9]:

- As a modulus within a prime elliptic curve: for example, the Mersenne prime $\left(2^{521}-1\right)$ is used to define an elliptic curve.

- In the Carter-Wegman Counter (CWC) mode [19], the Mersenne prime $\left(2^{127}-1\right)$ is used to define a universal hash function consisting of evaluating a polynomial modulo the Mersenne prime $\left(2^{127}-1\right)$.

In both cases, the special property that is taken advantage of is that Mersenne primes (rather than another prime of approximately the same size) make computing the modulo operation $x\mathrm{mod}\left(2^{521}-1\right)$ or $x\mathrm{mod}\left(2^{127}-1\right)$ easy by the linear-feedback shift register (LFSR). More generally, performing modular reduction modulo a Mersenne prime does not modify the hamming weight of the result.

On the other hand, in Asymmetric Key Cryptography, a pair of keys is used to encrypt and decrypt information. A receiver’s public key is used for encryption and a receiver’s private key is used for decryption. Public keys and Private keys are different. Even if the public key is known by everyone, the intended receiver can only decode it because he alone knows his private key. The most popular asymmetric key cryptography algorithm is the Rivest–Shamir–Adleman (RSA) algorithm [24]. The practical difficulty of factoring the product of two large prime numbers is what makes the RSA algorithm secure.

As seen, the number of Mersenne primes is relatively limited, and a fortiori, those of medium size are even less. As an alternative for asymmetric key cryptography, we propose to use Generalized Mersenne primes which are more frequent even for small prime exponents and for which both the base a and the exponent n can be used either as public keys or secret keys.

4. Conclusions

It was shown that with the proposed generalization of Mersenne numbers, for any natural integer base a, Generalized Mersenne numbers are in general such that $\left(G{M}_{a,n}-1\right)$ are even and divisible by n, a and $\left(a-1\right)$ for any odd prime exponent n and by $\left(a\left(a-1\right)+1\right)$ for any prime exponent $n>5$. The remaining factor is a function of triangular numbers of $\left(a-1\right)$, specific to each prime exponent n. Four theorems on Mersenne numbers were generalized for Generalized Mersenne numbers and four new theorems were demonstrated, allowing to show first, that $\left(G{M}_{a,n}-1\right)$ are divisible by 6, and more precisely $G{M}_{a,n}$ are congruent to $1\left(\mathrm{mod}12\right)$ or $7\left(\mathrm{mod}12\right)$ depending on the congruence of the base $a\left(mod4\right)$; second, that $\left(G{M}_{a,n}-1\right)$ are divisible by 10 if $n\equiv 1\left(\mathrm{mod}4\right)$ and, if $n\equiv 3\left(mod4\right)$, $G{M}_{a,n}\equiv 1\left(mod10\right)$, or $7\left(mod10\right)$ or $9\left(mod10\right)$ depending on the congruence of the base $a\left(mod5\right)$; third, that all factors $c_i$ of $G{M}_{a,n}$ are of the form $\left(2n{f}_i+1\right)$ with $f_i$ natural integers such that $c_i$ is prime itself or the product of primes of the form $\left(2nj+1\right)$ with j natural integer; fourth, that for odd prime exponents n, all $G{M}_{a,n}$ are periodically congruent to either $\pm 1\left(\mathrm{mod}8\right)$ or $\pm 3\left(\mathrm{mod}8\right)$ depending on the congruence of the base $a\left(mod8\right)$; and fifth, that the factors of a composite $G{M}_{a,n}$ is of the form $\left(2n{f}_i+1\right)$ with $f_i\equiv u\left(\mathrm{mod}4\right)$ and u being either 0, 1, 2 or 3 depending on the congruence of the exponent $n\left(mod4\right)$ and on the congruence of the base $a\left(mod8\right)$. Finally, the potential use of Generalized Mersenne primes in cryptography has been shortly addressed.

Distributions of primes and composites in Generalized Mersenne numbers are further investigated in two companion papers.