There Are Infinitely Many Sophie Germain Primes

1. Introduction

The search for a general formula to determine the $n^{th}$Sophie Germain prime is an ongoing challenge in mathematics. This paper produces a test for the Sophie Germain/Safe prime set. Sophie Germain primes $S_p$are generators of “Safe primes” , $P_{safe}=2P_s+1$ , where $P_S$is a Sophie prime. For generality, we will use $p$ when we refer to a prime. Not all primes $p,$ can generate a Safe prime . For example, the Sophie Germain primes, [3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953], are examples that generate Safe primes. It is extremely difficult to find the Sophie Germain primes $p$ without tedious factorization, since the known set of Sophie Germain primes, like the Mersenne primes are separated by long distances of non-conforming primes.

In this paper, a novel set of relations are developed for the Sophie Germain primes using the product Gamma-function, denoted as $\mathsf{\Gamma }\left(nx\right)$, first introduced by Swiss mathematician Leonhard Euler [1] 1729. Euler’s deep insights into $\mathsf{\Gamma }$-function led to numerous results that provide key insights into many fields of mathematics including Probability theory and Statistics. Other major contributions to the development of the $\mathsf{\Gamma }$-function used in this paper were developed by Carl Freidman Gauss [2]. Gauss’s work led to the famous reflection formula of the $\zeta$-function. A key insight into the $\mathsf{\Gamma }$-function is its multiplicative nature. New results will be presented in this paper resulting from the properties of the $\mathsf{\Gamma }$-function . So far, there has been little development in the additive representation of the $\mathsf{\Gamma }$-function as a series of simple terms.

The product-form of the $\mathsf{\Gamma }$-function due to Gauss, provides further insights into many relations that will be developed in this paper. The product form is given by, [4], p. 896:

$$\mathsf{\Gamma }\left(y·n\right)={\left(2\pi \right)}^{{\displaystyle \frac{1-y}{2}}}{y}^{\left(n·y\right)-{\displaystyle \frac{1}{2}}}\prod _{k=0}^{y-1}\mathsf{\Gamma }\left(n+{\displaystyle \frac{k}{y}}\right)\left(1.\right)$$

Certain invariant relations of the product $\mathsf{\Gamma }$-function will be developed in this paper to show the connections of the $\mathsf{\Gamma }$-function to other functions. The $\mathsf{\Gamma }$-function of Gauss is given in [4], p.1038. It is related to the $\zeta$-function reflection formula developed by

$$\mathsf{\Gamma }\left({\displaystyle \frac{s}{2}}\right){\pi }^{-{\displaystyle \frac{s}{2}}}\zeta \left(s\right)=\mathsf{\Gamma }\left({\displaystyle \frac{1-s}{2}}\right){\pi }^{{\displaystyle \frac{s-1}{2}}}\zeta \left(1-s\right)\left(1.\right)$$

Other relations exist that relate the two functions using Bernoulli numbers. These relations are well studied, and they provide a wealth of information in Number theory and many disciplines in Mathematics. In this article, I show new relations that govern Sophie Germain primes. All these special integer relations are connected in precious way by powers of $2\pi$.

2. Sophie Germain Numbers.

A Sophie Germain prime is a prime number $p$ for which $2p+1$ is also prime.

For example, the following set are Sophie Primes,$P_s.$ :

[3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953]. Their corresponding “Safe primes” :

[7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907]

These primes were named after Marie-Sophie Germain (1776–1831), one of the earliest prominent female mathematicians in modern Europe. Sophie Germain’s study of such primes emerged from her work on Fermat’s Last Theorem (FLT).

In 1820, she corresponded with Carl Friedrich Gauss and developed a method that became known as the Sophie Germain Criterion for FLT. Her theorem stated that if there exists a prime p such that no two nonzero $p^{th}$ power residues modulo $2p+1$ differ by 1, then Fermat’s equation $x^p+y^p=z^p$ has no nontrivial integer solutions. This result is one of the first partial proofs of FLT for infinitely many exponents. To construct such primes, Germain identified the condition $2p+1$ being prime, leading to the definition used today.

After Germain, her primes were studied sporadically, often in connection with residue properties and FLT.

20th Century: Interest revived in the context of cryptographic key generation, since both $p$ and $2p+1$ being prime yield strong cyclic groups for public-key systems (e.g., in Diffie–Hellman).

Modern research: It remains unproven whether infinitely many Sophie Germain primes exist, though it’s strongly conjectured — analogous to the Twin Prime Conjecture. The Hardy–Littlewood Conjecture (1923) predicts their asymptotic distribution:

$${\pi }_S\left(x\right)~2C_2{\displaystyle \frac{x}{{\left(\log x\right)}^2}}\left(1.\right)$$

where $C_2\approx 0.6601618$ is the twin-prime-constant. This connection is an aspect that I will explore as a diversion to Twin Prime, where if $p$ is a prime, then $p+2$ is also a prime.

Sophie Germain primes form the seed set of safe primes, which in turn are vital to cryptographic group theory.

They are also structurally related to twin primes and Mersenne primes, since each involves coupling primes through multiplicative or additive relations. My work, aims to show these are harmonic subclasses of a deeper $\sigma -tan$ lattice—suggesting an analytic reason for their infinitude. As with Mersenne primes, it is conjectured that there exist an infinite number of Sophie Germain primes.

3. The Invariance of the Gamma Function to Substitution $\mathit{\sigma }\left(\mathit{m}\right)\to \mathit{\sigma }\left(\mathit{m}+\mathbf{j}\right)$.

I first want to introduce the curious fact that any function with a relational product $\left\{n·y\right\},$can be represented by the Sums of Divisor function$,\sigma \left(m\right)$. Here is a simple example:

$$\log \left(n·y\right)=\log n+\log y,\left(1.\right)$$

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n},\mathrm{i}\mathrm{f}n·y=m,$ we can put $n=\sigma \left(m\right),y={\displaystyle \frac{m}{\sigma \left(m\right)}},$ and so,

$$\log \left(m\right)=\log \sigma \left(m\right)+\log {\displaystyle \frac{m}{\sigma \left(m\right)}}\left(1.\right)$$

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n},\mathrm{i}\mathrm{f}n·y=N_p,$ we can put $n=\sigma \left(N_p\right),y={\displaystyle \frac{N_p}{\sigma \left(N_p\right)}},$ then, a Perfect number $N_p,$ has the relation:

$$\log \left(N_p\right)=\log \left(\sigma \left(N_p\right)\right)+\log \left({\displaystyle \frac{N_p}{\sigma \left(N_p\right)}}\right)\left(1.\right)$$

$$\log \left(N_p\right)=\log \left(\sigma \left(N_p\right)\right)+\log \left({\displaystyle \frac{1}{2}}\right)\left(1.\right)$$

Here is another example:

If $n·y=m,$ we can put $n=\sigma \left(m\right),y={\displaystyle \frac{m}{\sigma \left(m\right)}},$and so, applied to the formula [3], p.41:

$$\begin{array}{c}\sin \left(n·x\right)=n\sin \left(x\right)\cos \left(x\right){\displaystyle \prod _{k=1}^{{\displaystyle \frac{n-2}{2}}}}\left(1-{\displaystyle \frac{{\sin }^2\left(x\right)}{{\sin }^2\left(\frac{k\pi }{n}\right)}}\right)\\ \cos \left(n·x\right)={\displaystyle \prod _{k=1}^{{\displaystyle \frac{n}{2}}}}\left(1-{\displaystyle \frac{{\sin }^2\left(x\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2n}\right)}}\right)\end{array},\left[n\mathrm{i}\mathrm{s}\mathrm{ }\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\right]\left(1.\right)$$

$$\begin{array}{c}\sin \left(n·x\right)=n\sin \left(x\right){\displaystyle \prod _{k=1}^{{\displaystyle \frac{n-1}{2}}}}\left(1-{\displaystyle \frac{{\sin }^2\left(x\right)}{{\sin }^2\left(\frac{k\pi }{n}\right)}}\right)\\ \cos \left(n·x\right)=\cos \left(x\right){\displaystyle \prod _{k=1}^{{\displaystyle \frac{n-1}{2}}}}\left(1-{\displaystyle \frac{{\sin }^2\left(x\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2n}\right)}}\right)\end{array},\left[n\mathrm{i}\mathrm{s}\mathrm{ }\mathrm{o}\mathrm{d}\mathrm{d}\right]\left(1.\right)$$

Interestingly, $\left(8\right)\in even,$ and $\left(9\right)\in odd$, differentiate between odd and even values of $n.$ Since primes have $\sigma \left(p\right)=p+1,$an even number, and $p+1$is always even except for the prime 2, the relations $\left(9\right)\in oddand$does not apply to primes! Since $\sigma \left(2\right)=3.$For example,

$$\cos \left(2\right)=\cos \left({\displaystyle \frac{2}{3}}\right)\prod _{k=1}^1\left(1-{\displaystyle \frac{{\sin }^2\left(\frac{2}{3}\right)}{{\sin }^2\frac{\left(2k-1\right)\pi }{6}}}\right),\left[\sigma \left(2\right)\mathrm{i}\mathrm{s}\mathrm{ }\mathrm{o}\mathrm{d}\mathrm{d}\right]\left(1.\right)$$

$$-0.4161468365\dots =0.7858872608..\left(1-{\displaystyle \frac{0.3823812134\dots }{0.2500000000}}\right)=-0.4161468365\dots \left(1.\right)$$

By using the sum of divisor function, for Perfect numbers, $N_p,$ the even trigonometric relations $\left[\left(8\right)\right]\in even,$ apply, but the relations, $\left[\left(9\right)\right]\in odd$ do not apply, so we can put in (8), $\sigma \left(N_p\right)=2N_p$. The fact that the sum of divisor function $\sigma \left(m\right),$can be manipulated this way leads to some interesting formulas that can produce significant and unexpected results. See [5].

4. Application of the Trigonometric Function to Sophie Germain Numbers.

A Sophie Germain primes $S_p,$ is defined as a number for which$S_p=2p+1.$ Consequently, since both numbers are primes, we have

$${\displaystyle \frac{\sigma \left(2p+1\right)}{\sigma \left(p\right)}}={\displaystyle \frac{2p+2}{p+1}}=2\left(1.\right)$$

This is very intimately related to Perfect numbers ${\displaystyle \frac{\sigma \left(N_p\right)}{N_p}}=2,$ were,

$$N_p\in \left\{6,\mathrm{ }28,\mathrm{ }496,\mathrm{ }8128,\mathrm{ }33550336,\mathrm{ }\mathrm{8589869056,137438691328},\mathrm{ }\mathrm{2305843008139952128,2658455991569831744654692615953842176},\dots \right\}$$

Hence for, example, in (8), putting $n=\sigma \left(j\right),\left(neven\right),x={\displaystyle \frac{1}{j}}:$ then, we have

$$\begin{array}{c}\sin \left({\displaystyle \frac{\sigma \left(j\right)}{j}}\right)=\sigma \left(j\right)\sin \left({\displaystyle \frac{1}{j}}\right)\cos \left({\displaystyle \frac{1}{j}}\right){\displaystyle \prod _{k=1}^{{\displaystyle \frac{\sigma \left(j\right)-2}{2}}}}\left(1-{\displaystyle \frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{k\pi }{\sigma \left(j\right)}\right)}}\right)\\ \cos \left({\displaystyle \frac{\sigma \left(j\right)}{j}}\right)={\displaystyle \prod _{k=1}^{{\displaystyle \frac{\sigma \left(j\right)}{2}}}}\left(1-{\displaystyle \frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(j\right)}\right)}}\right)\end{array},\left[\sigma \left(j\right)\mathrm{i}\mathrm{s}\mathrm{ }\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\right]\left(1.\right)$$

$$\tan \left({\displaystyle \frac{\sigma \left(j\right)}{j}}\right)={\displaystyle \frac{\sigma \left(j\right)\sin \left(\frac{1}{j}\right)\cos \left(\frac{1}{j}\right)\prod _{k=1}^{\frac{\sigma \left(j\right)-2}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{k\pi }{\sigma \left(j\right)}\right)}\right)}{\prod _{k=1}^{\frac{\sigma \left(j\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(j\right)}\right)}\right)}}\left[\sigma \left(j\right)\mathrm{i}\mathrm{s}\mathrm{ }\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\right]\left(1.\right)$$

$$\sigma \left(j\right)={\displaystyle \frac{\tan \left(\frac{\sigma \left(j\right)}{j}\right)\prod _{k=1}^{\frac{\sigma \left(j\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(j\right)}\right)}\right)}{\sin \left(\frac{1}{j}\right)\cos \left(\frac{1}{j}\right)\prod _{k=1}^{\frac{\sigma \left(j\right)-2}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{k\pi }{\sigma \left(j\right)}\right)}\right)}}\left[\sigma \left(j\right)\mathrm{i}\mathrm{s}\mathrm{ }\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\right]\left(1.\right)$$

LEMMA 1: The rational trigonometric functions $\sin \left({\displaystyle \frac{\sigma \left(j\right)}{j}}\right),\cos \left({\displaystyle \frac{\sigma \left(j\right)}{j}}\right)$ determine $PerfectNumbers,SophieGermainprimesandTwinPrimes.$

Proof: From (15),

$$\sigma \left(j\right)=2\left[{\displaystyle \frac{\tan \left(\frac{\sigma \left(j\right)}{j}\right)\prod _{k=1}^{\frac{\sigma \left(j\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(j\right)}\right)}\right)}{2\sin \left(\frac{1}{j}\right)\cos \left(\frac{1}{j}\right)\prod _{k=1}^{\frac{\sigma \left(j\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(j\right)}\right)}\right)}}\right]\left(1.\right)$$

$$\sigma \left(j\right)=2\left[{\displaystyle \frac{\tan \left(\frac{\sigma \left(j\right)}{j}\right)\prod _{k=1}^{\frac{\sigma \left(j\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(j\right)}\right)}\right)}{\sin \left(\frac{2}{j}\right)\prod _{k=1}^{\frac{\sigma \left(j\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{j}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(j\right)}\right)}\right)}}\right]\left(1.\right)$$

This is the relation for Perfect Number, and so we arrive at: If $j=$ $N_p$is a Perfect number, then, the equality applies only when. Since ${\displaystyle \frac{\sigma \left(N_p\right)}{N_p}}$, we have,

$$N_p={\displaystyle \frac{\tan \left(2\right)\prod _{k=1}^{\frac{\sigma \left(N_p\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{N_p}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(N_p\right)}\right)}\right)}{\sin \left(\frac{2}{N_p}\right)\left(\prod _{k=1}^{\frac{\sigma \left(N_p\right)}{2}-1}\left(1-\frac{{\sin }^2\left(\frac{1}{N_p}\right)}{{\sin }^2\left(\frac{k\pi }{\sigma \left(N_p\right)}\right)}\right)\right)}}\left(1.\right)$$

Hence the rational functions of the $\sigma$ function in the trigonometric functions encodes this behavior of various types of numbers classes.

For Sophie Germain primes, $p$we have:

$$\sigma \left(p\right)={\displaystyle \frac{\tan \left(\frac{\sigma \left(2p+1\right)}{\sigma \left(p\right)}\right)\prod _{k=1}^{\frac{\sigma \left(2p+1\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{\sigma \left(p\right)}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(2p+1\right)}\right)}\right)}{\sin \left(\frac{2}{\sigma \left(p\right)}\right)\left(\prod _{k=1}^{\frac{\sigma \left(2p+1\right)}{2}-1}\left(1-\frac{{\sin }^2\left(\frac{1}{\sigma \left(p\right)}\right)}{{\sin }^2\left(\frac{k\pi }{\sigma \left(2p+1\right)}\right)}\right)\right)}}\left(1.\right)$$

Then, since ${\displaystyle \frac{\sigma \left(2p+1\right)}{\sigma \left(p\right)}}=2,\sigma \left(p\right)={\displaystyle \frac{\tan \left(2\right)\prod _{k=1}^{\frac{\sigma \left(2p+1\right)}{2}}\left(1-\frac{{\sin }^2\left(\frac{1}{\sigma \left(p\right)}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(2p+1\right)}\right)}\right)}{\sin \left(\frac{2}{\sigma \left(p\right)}\right)\left(\prod _{k=1}^{\frac{\sigma \left(2p+1\right)}{2}-1}\left(1-\frac{{\sin }^2\left(\frac{1}{\sigma \left(p\right)}\right)}{{\sin }^2\left(\frac{k\pi }{\sigma \left(2p+1\right)}\right)}\right)\right)}},p\in S_p\left(1.\right)$

Equation (20) only holds for Sophie Germain Primes$p\in S_p$ . Hence, we find that (18) behave distinctly for the sets of Mersenne primes $p\in M_p$that yield Perfect numbers $N_p$, while (20) behaves distinctly for Sophie Germain primes $p\in S_p$. The connection between the two suggest that the infinitum condition applies equally to both sets of numbers if it applies to one or the other. Perfect numbers are dealt with in a yet unpublished paper “There are infinitely many Mersenne Primes”, MDPI: Manuscript ID:mathematics-3942588.

The relations (20) hold exclusively for all Sophie Germain Primes. The right-hand side of (20) depends on implicit rational relationships between $\sigma \left(p\right)$and $\sigma \left(2p+1\right).$ It is clear that the basic rational trigonometric functions capture the properties of special prime integers. We now explore the general forms of trigonometric and exponential forms that capture Sophie Germain Primes.

5. Structural Difference Between σ(p) and (p + 1) for Special Primes.

For a prime $p$, $\sigma \left(p\right)=p+1$. At first glance, $\sigma \left(p\right)$ and $p+1$ look identical for all mathematical operations. However, when $\sigma \left(p\right)$ is treated functionally inside a product operation, $\prod \left(f\right(p)$and in a sum operation $\sum f\left(p\right)$over primes, the distinction is that $\sigma \left(p\right)$ belongs to a multiplicative arithmetic function, while $p+1$ is just a linear term. Consider a trigonometric product expression such as:

$$\prod _p\sin \left({\displaystyle \frac{x}{\sigma \left(p\right)}}\right),\prod _p\sin \left({\displaystyle \frac{x}{p+1}}\right)$$

Although we know that $\sigma \left(p\right)=1+p$ is numerically correct for primes, when the $\sigma$ is inside a broader arithmetic context, especially if the product includes composite arguments or convolution terms, the product operation changes how the function expands when distributed through summation or logarithmic differentiation. That’s because σ satisfies, $\sigma \left(mn\right)=\sigma \left(m\right)\sigma \left(n\right)$ = for coprime $m,n$. The linear combination of $\left(p+1\right)$. does not extend multiplicatively. If a product expression depends on multiplicative structure, replacing $\sigma \left(p\right)$ with $\left(p+1\right)$ breaks that property and alters convergence, periodicity, or symmetry. Trigonometric functions like sin, cos, tan, are periodic and analytic, but when you plug in an arithmetic function, the periodicity couples to the multiplicative nature of $\sigma$. This behavior however is not true outside the fields of the product operation or the sum operation. This invariance in the product forms is highlighted by the following theorem due to the Author. See [5].

Theorem 1.

Let $f_j\left(z\right)>0,$represent one integer factor of $g\left(z\right)$, then, if,  $f_j\left(z\right)$ is not a number theoretic function,

$$\mathsf{\Gamma }\left(g\left(z\right)\right)={\left(2\pi \right)}^{{\displaystyle \frac{1-f_j\left(z\right)}{2}}}{\left(f_j\left(z\right)\right)}^{g\left(z\right)-{\displaystyle \frac{1}{2}}}\prod _{k=0}^{f_j\left(z\right)-1}\mathsf{\Gamma }\left({\displaystyle \frac{g\left(z\right)+k}{f_j\left(z\right)}}\right)\left(1.\right)$$

is invariant with respect to choices of any other factors of $g\left(z\right)$.

The significance of the Theorem 1, is its consequences for prime numbers, and their relations to functions like the$\zeta$-function and the sum of divisors function,$\sigma \left(m\right)$and primes.

• PROOF:
• Let ${\mathit{f}}_{\mathit{j}}\left(\mathit{z}\right),$ be some ${\mathit{j}}^{\mathit{t}\mathit{h}}$ integer factor of $\mathit{k}$-factors a real or complex function $\mathit{g}\left(\mathit{z}\right).$ Then,

  $$g\left(z\right)={\prod }_{n=1}^k{f}_n\left(z\right)=f_j\left(z\right)\left({\prod }_{n=1}^{j-1}{f}_n\left(z\right){\prod }_{n=j+1}^k{f}_n\left(z\right)\right)\left(1.\right)$$

The Gauss gamma product formula is a simple relation given by:

$$\mathsf{\Gamma }\left(n·y\right)={\left(2\pi \right)}^{{\displaystyle \frac{1-n}{2}}}{n}^{\left(n·y\right)-{\displaystyle \frac{1}{2}}}\prod _{k=0}^{n-1}\mathsf{\Gamma }\left(y+{\displaystyle \frac{k}{n}}\right)\left(1.\right)$$

Then, since $f_j\left(z\right),$ is an integer-factor of $g\left(z\right),$ we have, putting $n=f_j\left(z\right),$ $y={\prod }_{n=1}^{j-1}{f}_n\left(z\right){\prod }_{n=j+1}^k{f}_n\left(z\right)$ in

$g\left(z\right)=f_j\left(z\right)\left({\prod }_{n=1}^{j-1}{f}_n\left(z\right){\prod }_{n=j+1}^k{f}_n\left(z\right)\right)$. Then,

$$\mathsf{\Gamma }\left(g\left(z\right)\right)={\left(2\pi \right)}^{{\displaystyle \frac{1-f_j\left(z\right)}{2}}}{\left(f_j\left(z\right)\right)}^{g\left(z\right)-{\displaystyle \frac{1}{2}}}\prod _{k=0}^{f_j\left(z\right)-1}\mathsf{\Gamma }\left({\displaystyle \frac{g\left(z\right)+k}{f_j\left(z\right)}}\right)\left(1.\right)$$

If there any other integer factor labelled here $f_v\left(z\right),\in Z,$then, the substitution $f_j\left(z\right)\to f_v\left(z\right)$ leaves $\mathsf{\Gamma }\left(g\left(z\right)\right)$ invariant. However, this is also true for any integer factors, $m$ of $f_v\left(z\right),$ then, for any $m,$

$$\mathsf{\Gamma }\left(g\left(z\right)\right)={\left(2\pi \right)}^{{\displaystyle \frac{1-m}{2}}}{\left(m\right)}^{g\left(z\right)-{\displaystyle \frac{1}{2}}}\prod _{k=0}^{m-1}\mathsf{\Gamma }\left({\displaystyle \frac{g\left(z\right)+k}{m}}\right)\left(1.\right)$$

remains invariant to the substitutions$f_j\left(z\right)\to f_v\left(z\right)\to m.$

It is clear that inside the product terms, we have a different set of rules. Hence the Corollary applies in the sense that if $f_j\left(z\right)$ is a non-integer number theoretic function, then the invariance does not apply. However, when we consider rational functions outside of the product, we find that simple arithmetic operations do apply, hence, (20).

6. The Relation of the Product Gamma Function to Primes.

From the Gauss $\mathsf{\Gamma }$-product formula,

$$\mathsf{\Gamma }\left(m\right)={\left(2\pi \right)}^{{\displaystyle \frac{1-\sigma \left(m\right)}{2}}}{\sigma \left(m\right)}^{m-{\displaystyle \frac{1}{2}}}\prod _{k=0}^{\sigma \left(m\right)-1}\mathsf{\Gamma }\left({\displaystyle \frac{m+k}{\sigma \left(m\right)}}\right)\left(1.\right)$$

It is clear that the following relations are equal,

$${\left(2\pi \right)}^{{\displaystyle \frac{1-m}{2}}}{m}^{m-{\displaystyle \frac{1}{2}}}\prod _{k=0}^{m-1}\mathsf{\Gamma }\left({\displaystyle \frac{m+k}{m}}\right)={\left(2\pi \right)}^{{\displaystyle \frac{1-\sigma \left(m\right)}{2}}}{\sigma \left(m\right)}^{m-{\displaystyle \frac{1}{2}}}\prod _{k=0}^{\sigma \left(m\right)-1}\mathsf{\Gamma }\left({\displaystyle \frac{m+k}{\sigma \left(m\right)}}\right)\left(1.\right)$$

Then, for all real numbers $m$,

$${\left(2\pi \right)}^{{\displaystyle \frac{\sigma \left(m\right)-m}{2}}}{\left({\displaystyle \frac{m}{\sigma \left(m\right)}}\right)}^{m-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{\prod _{k=0}^{m-1}\mathsf{\Gamma }\left(\frac{m+k}{m}\right)}{\prod _{k=0}^{\sigma \left(m\right)-1}\mathsf{\Gamma }\left(\frac{m+k}{\sigma \left(m\right)}\right)}}=1\left(1.\right)$$

Since, for all primes, $m=p$, $\sigma \left(p\right)=p+1,$ from (28), we get for all primes, $p$:

$$\sqrt{\left(2\pi \right)}{\displaystyle \frac{p^{p-\frac{1}{2}}\prod _{k=0}^{p-1}\mathsf{\Gamma }\left(\frac{p+k}{p}\right)}{{\sigma \left(p\right)}^{p-\frac{1}{2}}\prod _{k=0}^{\sigma \left(p\right)-1}\mathsf{\Gamma }\left(\frac{p+k}{\sigma \left(p\right)}\right)}}=1\left(1.\right)$$

and so, (29) is not invariant to the substitutions, $\sigma \left(p\right)\to \sigma \left(p+j\right)$, unless $\left\{p,p+j\right\}\in primes.$

7. The General Relation That Captures the Behavior of Sophie Germain Primes.

A Sophie Germain Prime, $\mathit{p}$, generates another prime, $2\mathit{p}+1$. Using (29) for both primes we get:

$$\left\{\sqrt{\left(2\pi \right)}{\displaystyle \frac{p^{p-\frac{1}{2}}\prod _{k=0}^{p-1}\mathsf{\Gamma }\left(\frac{p+k}{p}\right)}{{\sigma \left(p\right)}^{p-\frac{1}{2}}\prod _{k=0}^{\sigma \left(p\right)-1}\mathsf{\Gamma }\left(\frac{p+k}{\sigma \left(p\right)}\right)}}\right\}\left\{\sqrt{\left(2\pi \right)}{\displaystyle \frac{{\left(2p+1\right)}^{2p+1-\frac{1}{2}}\prod _{k=0}^{2p+1-1}\mathsf{\Gamma }\left(\frac{2p+1+k}{2p+1}\right)}{{\sigma \left(2p+1\right)}^{2p+1-\frac{1}{2}}\prod _{k=0}^{\sigma \left(2p+1\right)-1}\mathsf{\Gamma }\left(\frac{2p+1+k}{\sigma \left(p\right)}\right)}}\right\}=1\left(1.\right)$$

$$\left\{\left(2\mathit{\pi }\right)\mathit{ }{\left({\displaystyle \frac{\mathit{p}}{\mathit{\sigma }\left(\mathit{p}\right)}}\right)}^{\mathit{p}-{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{2\mathit{p}+1}{\mathit{\sigma }\left(2\mathit{p}+1\right)}}\right)}^{2\mathit{p}+{\displaystyle \frac{1}{2}}}\right\}\mathit{ }\left\{\mathit{ }{\displaystyle \frac{\prod _{\mathit{k}=0}^{\mathit{p}-1}\mathsf{\Gamma }\left(\frac{\mathit{p}+\mathit{k}}{\mathit{p}}\right)\prod _{\mathit{k}=0}^{2\mathit{p}}\mathsf{\Gamma }\left(\frac{2\mathit{p}+1+\mathit{k}}{2\mathit{p}+1}\right)}{\prod _{\mathit{k}=0}^{\mathit{\sigma }\left(\mathit{p}\right)-1}\mathsf{\Gamma }\left(\frac{\mathit{p}+\mathit{k}}{\mathit{\sigma }\left(\mathit{p}\right)}\right)\prod _{\mathit{k}=0}^{\mathit{\sigma }\left(2\mathit{p}+1\right)-1}\mathsf{\Gamma }\left(\frac{2\mathit{p}+1+\mathit{k}}{\mathit{\sigma }\left(\mathit{p}\right)}\right)}}\right\}=1\mathit{ }\left(1.\right)$$

Lemma 1: Analytic Reduction for Sophie Germain Primes Pairs.

Let $\mathit{p}$ be a prime with $\mathit{q}=2\mathit{p}+1,$ a prime also, then (31) holds. To prove this, see SECTION 9 above using the Weierstrass-Euler Expansion to determine “primeness”. Relation (31) holds true for all Sophie primes sets $\left\{\mathit{p},2\mathit{p}+1\right\}$.

Now in general, for any $\mathit{n}$, we set:

$$\left\{\left(2\mathit{\pi }\right)\mathit{ }{\left({\displaystyle \frac{\mathit{n}}{\mathit{\sigma }\left(\mathit{n}\right)}}\right)}^{\mathit{n}-{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{2\mathit{n}+1}{\mathit{\sigma }\left(2\mathit{n}+1\right)}}\right)}^{2\mathit{n}+{\displaystyle \frac{1}{2}}}\right\}\mathit{ }\left\{\mathit{ }{\displaystyle \frac{\prod _{\mathit{k}=0}^{\mathit{n}-1}\mathsf{\Gamma }\left(\frac{\mathit{n}+\mathit{k}}{\mathit{n}}\right)\prod _{\mathit{k}=0}^{2\mathit{n}}\mathsf{\Gamma }\left(\frac{2\mathit{n}+1+\mathit{k}}{2\mathit{n}+1}\right)}{\prod _{\mathit{k}=0}^{\mathit{\sigma }\left(\mathit{n}\right)-1}\mathsf{\Gamma }\left(\frac{\mathit{n}+\mathit{k}}{\mathit{\sigma }\left(\mathit{n}\right)}\right)\prod _{\mathit{k}=0}^{\mathit{\sigma }\left(2\mathit{n}+1\right)-1}\mathsf{\Gamma }\left(\frac{2\mathit{n}+1+\mathit{k}}{\mathit{\sigma }\left(\mathit{n}\right)}\right)}}\right\}={\left(2\mathit{\pi }\right)}^{-{\mathit{a}}_{\mathit{n}}}\mathit{ }\left(1.\right)$$

The values of ${\mathit{a}}_{\mathit{n}}={\mathit{M}}_{\mathit{n}}$, for $\mathit{n}=2\dots 100,$ are shown in Table 1 below, with ${\mathit{a}}_{\mathit{n}}=0,$ only for Sophie primes.

It is clear that in (32) , ${\mathit{a}}_{\mathit{n}}$=0, for the Sophie Germain primes :

[3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953].

Hence the dual prime condition gives us the correct result Sophie Germain primes in (30).

The Plot in Figure 1 shows the values of ${\mathit{a}}_{\mathit{n}}\mathbf{v}\mathbf{e}\mathbf{r}\mathbf{s}\mathbf{u}\mathbf{s}\mathit{n}=2\dots 500.$

It is clear that the average trend for ${\mathit{a}}_{\mathit{n}}$ increases with increasing $\mathit{n}.$

The Sophie relation :

$$\mathit{\sigma }\left(\mathit{p}\right)={\displaystyle \frac{\mathbf{tan}\left(2\right)\prod _{\mathit{k}=1}^{\frac{\mathit{\sigma }\left(2\mathit{p}+1\right)}{2}}\left(1-\frac{{\mathbf{sin}}^2\left(\frac{1}{\mathit{\sigma }\left(\mathit{p}\right)}\right)}{{\mathbf{sin}}^2\left(\frac{\left(2\mathit{k}-1\right)\mathit{\pi }}{2\mathit{\sigma }\left(2\mathit{p}+1\right)}\right)}\right)}{\mathit{ }\mathbf{sin}\left(\frac{2}{\mathit{\sigma }\left(\mathit{p}\right)}\right)\left(\prod _{\mathit{k}=1}^{\frac{\mathit{\sigma }\left(2\mathit{p}+1\right)}{2}-1}\left(1-\frac{{\mathbf{sin}}^2\left(\frac{1}{\mathit{\sigma }\left(\mathit{p}\right)}\right)}{{\mathbf{sin}}^2\left(\frac{\mathit{k}\mathit{\pi }}{\mathit{\sigma }\left(2\mathit{p}+1\right)}\right)}\right)\right)}}\mathit{ },\mathit{p}\in {\mathit{P}}_{\mathit{S}}\mathit{ }\left(1.\right)$$

is analogous to the Perfect number relation:

$$N_p={\displaystyle \frac{\tan \left(2\right)\prod _{k=1}^{N_p}\left(1-\frac{{\sin }^2\left(\frac{1}{N_p}\right)}{{\sin }^2\left(\frac{\left(2k-1\right)\pi }{2\sigma \left(N_p\right)}\right)}\right)}{\sin \left(\frac{2}{N_p}\right)\left(\prod _{k=1}^{N_p-1}\left(1-\frac{{\sin }^2\left(\frac{1}{N_p}\right)}{{\sin }^2\left(\frac{k\pi }{\sigma \left(N_p\right)}\right)}\right)\right)}},p\in P_M\left(1.\right)$$

This allows us to define a new sort of Sophie Perfect number ${\mathit{N}}_{\mathit{p}}\to {\mathit{N}}_{\mathit{S}}=\mathit{\sigma }\left(\mathit{p}\right),$ with the relation (33) and (34). However, the product formulation is not invariant to the substitution of $\mathit{p}\to 2\mathit{p}+1,$ in (33) due to internal cancellations in the ${\mathbf{s}\mathbf{i}\mathbf{n}}^2$ terms, and further the coupling $\mathit{\sigma }\left(2\mathit{p}+1\right)\to \mathit{\sigma }\left(4\mathit{p}+3\right)$, destroys the definition of the “Safe Prime”$,2\mathit{p}+1$, since $4\mathit{p}+3$ is not a safe prime except for certain values of $\mathit{p}.$ Hence we need the proper transformation to make the substitution: $\mathit{\sigma }\left(2\mathit{p}+1\right)=2\mathit{\sigma }\left(\mathit{p}\right)$ in (33).

The invariance of (34) to the substitution $\mathit{\sigma }\left(2\mathit{p}+1\right)=2\mathit{\sigma }\left(\mathit{p}\right)$, is a key property of the coupling invariance allowed for the unique relation for tan(2).

$${\mathit{N}}_{\mathit{S}}={\displaystyle \frac{\mathbf{tan}\left(2\right)\prod _{\mathit{k}=1}^{{\mathit{N}}_{\mathit{S}}}\left(1-\frac{{\mathbf{sin}}^2\left(\frac{1}{{\mathit{N}}_{\mathit{S}}}\right)}{{\mathbf{sin}}^2\left(\frac{\left(2\mathit{k}-1\right)\mathit{\pi }}{4{\mathit{N}}_{\mathit{S}}}\right)}\right)}{\mathit{ }\mathbf{sin}\left(\frac{2}{{\mathit{N}}_{\mathit{S}}}\right)\left(\prod _{\mathit{k}=1}^{{\mathit{N}}_{\mathit{S}}-1}\left(1-\frac{{\mathbf{sin}}^2\left(\frac{1}{{\mathit{N}}_{\mathit{S}}}\right)}{{\mathbf{sin}}^2\left(\frac{\mathit{k}\mathit{\pi }}{{2\mathit{N}}_{\mathit{S}}}\right)}\right)\right)}}\mathit{ },\mathit{ }\mathit{p}\in {\mathit{P}}_{\mathit{S}}\mathit{ }\left(1.\right)$$

In-fact the (35) gives exact results when ${\mathit{N}}_{\mathit{S}}=\mathit{\sigma }\left(\mathit{p}\right).$

$$\mathit{\sigma }\left(\mathit{p}\right)={\displaystyle \frac{\mathbf{tan}\left(2\right)\prod _{\mathit{k}=1}^{\mathit{\sigma }\left(\mathit{p}\right)}\left(1-\frac{{\mathbf{sin}}^2\left(\frac{1}{\mathit{\sigma }\left(\mathit{p}\right)}\right)}{{\mathbf{sin}}^2\left(\frac{\left(2\mathit{k}-1\right)\mathit{\pi }}{4\mathit{\sigma }\left(\mathit{p}\right)}\right)}\right)}{\mathit{ }\mathbf{sin}\left(\frac{2}{\mathit{\sigma }\left(\mathit{p}\right)}\right)\left(\prod _{\mathit{k}=1}^{\mathit{\sigma }\left(\mathit{p}\right)-1}\left(1-\frac{{\mathbf{sin}}^2\left(\frac{1}{\mathit{\sigma }\left(\mathit{p}\right)}\right)}{{\mathbf{sin}}^2\left(\frac{\mathit{k}\mathit{\pi }}{2\mathit{\sigma }\left(\mathit{p}\right)}\right)}\right)\right)}}\mathit{ },\mathit{p}\in {\mathit{P}}_{\mathit{S}}\mathit{ }\left(1.\right)$$

Hence in the Sophie world, a phase doubling occurs replacing the shift, $\mathit{\sigma }\left(2\mathit{p}+1\right)=2\mathit{\sigma }\left(\mathit{p}\right)$ is a Sophie perfect number ${\mathit{N}}_{\mathit{S}},$ if $\mathit{p}\in {\mathit{P}}_{\mathit{S}}.$

Defining a Sophie Perfect Number ${\mathit{N}}_{\mathit{S}}$.

Let $\mathit{\sigma }\left({\mathit{N}}_{\mathit{S}}\right)=\left(2\mathit{p}+1\right){\mathit{N}}_{\mathit{S}}$ for some prime $\mathit{p}$ that also satisfies $2\mathit{p}+1$ a prime. Then, ${\displaystyle \frac{\mathit{\sigma }\left({\mathit{N}}_{\mathit{S}}\right)}{{\mathit{N}}_{\mathit{S}}}}=2\mathit{p}+1.$ In classical number theory, an integer $\mathit{n}$ is perfect if $\mathit{\sigma }\left(\mathit{n}\right)=2\mathit{n}.$ This relation expresses an eigenvalue equation under the multiplicative operator

$$\mathcal{M}f={\displaystyle \frac{\sigma \left(n\right)}{n}}=2$$

Perfect numbers thus correspond to the fixed points of a multiplicative scaling by 2. Sophie Germain primes introduce an additive and multiplicative relationship between primes $\left\{p,q\right\}$: $q=2p+1.$ This reveals an arithmetic 2:1 resonance between $\sigma \left(p\right)$ and $\sigma \left(q\right).$ The analogue of a Sophie perfect number $N_S$, and a Perfect Number $N_{p\in Mersenneprimes}$can be summarized in the following Table 2:

Hence, the Sophie Transform corresponds to a phase-doubling trigonometric transform—a discrete analogue of the Fourier dilation symmetry $F\left(2x\right)=2F(x$). Perfect numbers represent amplitude-doubling; Sophie-perfect numbers represent frequency-doubling within the divisor-sum spectrum.