On a Conjecture Involving Twin Practical Numbers

1. Introduction

A positive integer m is called a practical number if each $n=1,\dots ,m$ can be written as the sum of some distinct divisors of m. Clearly all practical numbers are even except 1, and all even perfect numbers and powers of 2 are practical. Let $P\left(x\right)$ denote the number of practical numbers less than x. In 1984, Hausman and Shapiro [1] showed that

$$P\left(x\right)\ll {\displaystyle \frac{x}{{(logx)}^{\beta }}}$$

(1)

for any $\beta <{\displaystyle \frac{1}{2}}{(1-{\displaystyle \frac{1}{log2}})}^2\approx 0.0979$. Margenstern [2] showed in 1991 that

$$P\left(x\right)\gg {\displaystyle \frac{x}{exp\left(\frac{1}{2log2}{(loglogx)}^2+3loglogx\right)}}.$$

(2)

Tenenbaum [3,4] improved the bounds (1) and (2) to

$${\displaystyle \frac{x}{logx}}{(loglogx)}^{-{\displaystyle \frac{5}{3}}-\epsilon }\ll P\left(x\right)\ll {\displaystyle \frac{x}{logx}}loglogxlogloglogx.$$

(3)

Using a sieve method, Saias [5] improved Tenenbaum’s result to

$${\displaystyle \frac{x}{logx}}\ll P\left(x\right)\ll {\displaystyle \frac{x}{logx}}.$$

(4)

Margenstern conjectured that

$$P\left(x\right)\sim {\lambda }_1{\displaystyle \frac{x}{logx}}$$

(5)

with ${\lambda }_1\approx 1.341$.

Like the famous twin prime conjecture, Margenstern [2] established that there are infinitely many twin practical numbers m and $m+2$, and another detailed proof was given by Melfi [6] in 1996. Let $P_2\left(x\right)$ denote the number of practical number pairs $(m,m+2)$ less than x. Margenstern’s proof leads to a lower bound $P_2\left(x\right)\gg loglogx$, and Melfi’s proof leads to a better lower bound $P_2\left(x\right)\gg logx$. In 2002, Melfi [7] improved the lower bound to

$$P_2\left(x\right)\gg {\displaystyle \frac{x}{exp\left(k{(logx)}^{\frac{1}{2}}\right)}}$$

(6)

with $k>2+log{\displaystyle \frac{3}{2}}$. Margenstern also conjectured that

$$P_2\left(x\right)\sim {\lambda }_2{\displaystyle \frac{x}{logx}}$$

(7)

with ${\lambda }_1\approx 1.436$.

In 2022, Wang and Sun [8] generalized Margenstern’s result and showed that there are infinitely many practical numbers q such that $q^4$ and $q^4+2$ are also practical numbers. They showed their result by modifying Melfi’s cyclotomic method.

In this paper, we further state a conjecture and show some new results. For any positive integer C, there are infinitely many practical numbers q such that $q^C$ and $q^C+2$ are also practical numbers. Clearly the results of Margenstern [2] and Melfi [6,7] imply the case $C=1$ of Conjecture Section 1, and Wang and Sun [8] solved the cases $C=2$ and $C=4$. Wang and Sun [8] solved the cases $C=2$ and $C=4$ by proving that the sequence $2^{35\times 3^k+1}+2$ are practical for every integer $k⩾0$. In this paper, we find four new sequences that generate infinitely many practical numbers.

Theorem 1. 

$2^{135\times 7^k+1}+2$, $2^{165\times 3^k+1}+2$, $2^{175\times 3^k+1}+2$ and $2^{189\times 5^k+1}+2$ are practical for every integer $k⩾0$.

Using Theorem 1 we can easily deduce the following three Theorems.

Theorem 2. 

Let p denotes a prime number, $7⩽p<100$. Then there are infinitely many practical numbers q such that $q^p$ and $q^p+2$ are also practical numbers.

Theorem 3. 

There are infinitely many practical numbers q such that:

(1). $q^3$ and $q^3+8$ are also practical numbers.

(2). $q^5$ and $q^5+32$ are also practical numbers.

Theorem 4. 

For any positive integer C, there exists an integer h such that there are infinitely many practical numbers q such that $q^C$ and $q^C+2^h$ are also practical numbers.

2. Proof of Theorem 1.2

In order to prove our Theorems, we need the following structure theorem:

Lemma 1 

([6], Lemma 1). Let m be any practical number. Then $mn$ is practical for every $n=1,\dots ,\sigma \left(m\right)+1$. In particular, $mn$ is practical for every $1⩽n⩽2m$.

Let ${\Phi }_m\left(x\right)$ denotes the mth cyclotomic polynomial, then we have

Lemma 2. 

$$x^n-1=\prod _{d\mid n}{\Phi }_d\left(x\right)\mathrm{for}\mathrm{all}n=1,2,3,\dots$$

Now we start our proof of Theorem 1. Note that we may use $m_k$ to represent different sequences in different subsection.

2.1. Sequence 1

We shall prove that $2^{135\times 7^k+1}+2$ are practical for every $k⩾0$.

Write $m_k=2^{135\times 7^k+1}+2$ for $k=0,1,2,\dots$, then we only need to show that $m_k$ is practical for every $k=0,1,2,\dots$. By a computer calculation we find that

$$\begin{array}{cc}\hfill m_0=& 87112285931760246646623899502532662132738\hfill \\ \hfill =& 2\times 3^4\times 11\times 19\times 331\times 811\times 15121\times 87211\times 18837001\times 385838642647891.\hfill \end{array}$$

By Lemma 1, we know that $m_0$ is practical. Let $x=2^{7^k}$, then we have $m_k=2(x^{135}+1)$ and $m_{k+1}=2(x^{945}+1)$. Since $k⩾0$, we have $x⩾2$. By Lemma 2, we have

$${\displaystyle \frac{x^{1890}+1}{x^{945}+1}}={\displaystyle \frac{x^{270}+1}{x^{135}+1}}{\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{378}\left(x\right){\Phi }_{630}\left(x\right){\Phi }_{1890}\left(x\right).$$

(8)

Assume that $2(x^{135}+1)$ is practical, then we only need to show that

$$2(x^{135}+1){\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{378}\left(x\right){\Phi }_{630}\left(x\right){\Phi }_{1890}\left(x\right)$$

is practical.

Note that

$$\begin{array}{cc}\hfill {\Phi }_{14}\left(x\right)=& x^6-x^5+x^4-x^3+x^2-x+1,\hfill \\ \hfill {\Phi }_{42}\left(x\right)=& x^{12}+x^{11}-x^9-x^8+x^6-x^4-x^3+x+1,\hfill \\ \hfill {\Phi }_{70}\left(x\right)=& x^{24}+x^{23}-x^{19}-x^{18}-x^{17}-x^{16}+x^{14}+x^{13}+x^{12}+x^{11}\hfill \\ & +x^{10}-x^8-x^7-x^6-x^5+x+1,\hfill \\ \hfill {\Phi }_{126}\left(x\right)=& x^{36}+x^{33}-x^{27}-x^{24}+x^{18}-x^{12}-x^9+x^3+1,\hfill \\ \hfill {\Phi }_{210}\left(x\right)=& x^{48}-x^{47}+x^{46}+x^{43}-x^{42}+2x^{41}-x^{40}+x^{39}+x^{36}-x^{35}\hfill \\ & +x^{34}-x^{33}+x^{32}-x^{31}-x^{28}-x^{26}-x^{24}-x^{22}-x^{20}-x^{17}\hfill \\ & +x^{16}-x^{15}+x^{14}-x^{13}+x^{12}+x^9-x^8\hfill \\ & +2x^7-x^6+x^5+x^2-x+1,\hfill \\ \hfill {\Phi }_{378}\left(x\right)=& x^{108}+x^{99}-x^{81}-x^{72}+x^{54}-x^{36}-x^{27}+x^9+1,\hfill \\ \hfill {\Phi }_{630}\left(x\right)=& x^{144}-x^{141}+x^{138}+x^{129}-x^{126}+2x^{123}-x^{120}\hfill \\ & +x^{117}+x^{108}-x^{105}+x^{102}-x^{99}+x^{96}-x^{93}-x^{84}\hfill \\ & -x^{78}-x^{72}-x^{66}-x^{60}-x^{51}+x^{48}-x^{45}+x^{42}\hfill \\ & -x^{39}+x^{36}+x^{27}-x^{24}+2x^{21}-x^{18}+x^{15}+x^6-x^3+1,\hfill \\ \hfill {\Phi }_{1890}\left(x\right)=& x^{432}-x^{423}+x^{414}+x^{387}-x^{378}+2x^{369}-x^{360}+x^{351}\hfill \\ & +x^{324}-x^{315}+x^{306}-x^{297}+x^{288}-x^{279}-x^{252}-x^{234}\hfill \\ & -x^{216}-x^{198}-x^{180}-x^{153}+x^{144}-x^{135}+x^{126}-x^{117}\hfill \\ & +x^{108}+x^{81}-x^{72}+2x^{63}-x^{54}+x^{45}+x^{18}-x^9+1.\hfill \end{array}$$

Since $x⩾2$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^6⩽& {\Phi }_{14}\left(x\right)⩽x^6,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill x^{12}⩽& {\Phi }_{42}\left(x\right)⩽2x^{12},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill x^{24}⩽& {\Phi }_{70}\left(x\right)⩽2x^{24},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill x^{36}⩽& {\Phi }_{126}\left(x\right)⩽2x^{36},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{48}⩽& {\Phi }_{210}\left(x\right)⩽x^{48},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill x^{108}⩽& {\Phi }_{378}\left(x\right)⩽2x^{108},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{144}⩽& {\Phi }_{630}\left(x\right)⩽x^{144},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{432}⩽& {\Phi }_{1890}\left(x\right)⩽x^{432}.\hfill \end{array}$$

(16)

By (9)–(13), we have

$${\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right)⩽8x^{126}⩽x^{129}⩽2(x^{135}+1)$$

(17)

and

$$2(x^{135}+1){\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right)$$

is practical by Lemma 1.

By (9)–(13), we have

$$2(x^{135}+1){\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right)⩾{\displaystyle \frac{1}{2}}{x}^{261}⩾x^{260}.$$

(18)

By (14)–(15), we have

$${\Phi }_{378}\left(x\right){\Phi }_{630}\left(x\right)⩽2x^{252}⩽x^{253}.$$

(19)

By (18)–(19), we have

$${\Phi }_{378}\left(x\right){\Phi }_{630}\left(x\right)⩽x^{253}⩽x^{260}⩽2(x^{135}+1){\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right)$$

(20)

and

$$2(x^{135}+1){\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{378}\left(x\right){\Phi }_{630}\left(x\right)$$

is practical by Lemma 1.

By (9)–(15), we have

$$2(x^{135}+1){\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{378}\left(x\right){\Phi }_{630}\left(x\right)⩾{\displaystyle \frac{1}{4}}{x}^{513}⩾x^{511}.$$

(21)

By (16) and (21), we have

$${\Phi }_{1890}\left(x\right)⩽x^{432}⩽x^{511}⩽2(x^{135}+1){\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{378}\left(x\right){\Phi }_{630}\left(x\right)$$

(22)

and

$$2(x^{135}+1){\Phi }_{14}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{126}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{378}\left(x\right){\Phi }_{630}\left(x\right){\Phi }_{1890}\left(x\right)$$

is practical by Lemma 1.

Since $m_0$ is practical, we can show that $m_k$ is practical for every $k=0,1,2,\dots$ by an inductive process. Now the proof of sequence $2^{135\times 7^k+1}+2$ is completed.

2.2. Sequence 2

We shall prove that $2^{165\times 3^k+1}+2$ are practical for every $k⩾0$.

Write $m_k=2^{165\times 3^k+1}+2$ for $k=0,1,2,\dots$, then we only need to show that $m_k$ is practical for every $k=0,1,2,\dots$. By a computer calculation we find that

$$\begin{array}{cc}\hfill m_0=& 93536104789177786765035829293842113257979682750466\hfill \\ \hfill =& 2\times 3^2\times {11}^2\times 67\times 331\times 683\times 2971\times 20857\times 48912491\hfill \\ & \times 415365721\times 2252127523412251.\hfill \end{array}$$

By Lemma 1, we know that $m_0$ is practical. Let $x=2^{3^k}$, then we have $m_k=2(x^{165}+1)$ and $m_{k+1}=2(x^{495}+1)$. Since $k⩾0$, we have $x⩾2$. By Lemma 2, we have

$${\displaystyle \frac{x^{990}+1}{x^{495}+1}}={\displaystyle \frac{x^{330}+1}{x^{165}+1}}{\Phi }_{18}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{198}\left(x\right){\Phi }_{990}\left(x\right).$$

(23)

Assume that $2(x^{165}+1)$ is practical, then we only need to show that

$$2(x^{165}+1){\Phi }_{18}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{198}\left(x\right){\Phi }_{990}\left(x\right)$$

is practical.

Note that

$$\begin{array}{cc}\hfill {\Phi }_{18}\left(x\right)=& x^6-x^3+1,\hfill \\ \hfill {\Phi }_{90}\left(x\right)=& x^{24}+x^{21}-x^{15}-x^{12}-x^9+x^3+1,\hfill \\ \hfill {\Phi }_{198}\left(x\right)=& x^{60}+x^{57}-x^{51}-x^{48}+x^{42}+x^{39}-x^{33}\hfill \\ & -x^{30}-x^{27}+x^{21}+x^{18}-x^{12}-x^9+x^3+1,\hfill \\ \hfill {\Phi }_{990}\left(x\right)=& x^{240}-x^{237}+x^{234}+x^{225}-x^{222}+x^{219}\hfill \\ & +x^{207}-x^{204}+x^{201}-x^{195}+2x^{192}-2x^{189}\hfill \\ & +x^{186}-x^{180}+x^{177}-x^{174}-x^{162}+x^{159}\hfill \\ & -x^{156}+x^{150}-2x^{147}+2x^{144}-2x^{141}+x^{138}\hfill \\ & -x^{132}+x^{129}-x^{126}+x^{123}-x^{120}+x^{117}\hfill \\ & -x^{114}+x^{111}-x^{108}+x^{102}-2x^{99}+2x^{96}\hfill \\ & -2x^{93}+x^{90}-x^{84}+x^{81}-x^{78}-x^{66}+x^{63}\hfill \\ & -x^{60}+x^{54}-2x^{51}+2x^{48}-x^{45}+x^{39}-x^{36}\hfill \\ & +x^{33}+x^{21}-x^{18}+x^{15}+x^6-x^3+1.\hfill \end{array}$$

Since $x⩾2$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^6⩽& {\Phi }_{18}\left(x\right)⩽x^6,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill x^{24}⩽& {\Phi }_{90}\left(x\right)⩽2x^{24},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill x^{60}⩽& {\Phi }_{198}\left(x\right)⩽2x^{60},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{240}⩽& {\Phi }_{990}\left(x\right)⩽x^{240}.\hfill \end{array}$$

(27)

By (24)–(26), we have

$${\Phi }_{18}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{198}\left(x\right)⩽4x^{90}⩽x^{92}⩽2(x^{165}+1)$$

(28)

and

$$2(x^{165}+1){\Phi }_{18}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{198}\left(x\right)$$

is practical by Lemma 1.

By (24)–(26), we have

$$2(x^{165}+1){\Phi }_{18}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{198}\left(x\right)⩾x^{255}.$$

(29)

By (27) and (29), we have

$${\Phi }_{990}\left(x\right)⩽x^{240}⩽x^{255}⩽2(x^{165}+1){\Phi }_{18}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{198}\left(x\right)$$

(30)

and

$$2(x^{165}+1){\Phi }_{18}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{198}\left(x\right){\Phi }_{990}\left(x\right)$$

is practical by Lemma 1.

Since $m_0$ is practical, we can show that $m_k$ is practical for every $k=0,1,2,\dots$ by an inductive process. Now the proof of sequence $2^{165\times 3^k+1}+2$ is completed.

2.3. Sequence 3

We shall prove that $2^{175\times 3^k+1}+2$ are practical for every $k⩾0$.

Write $m_k=2^{175\times 3^k+1}+2$ for $k=0,1,2,\dots$, then we only need to show that $m_k$ is practical for every $k=0,1,2,\dots$. By a computer calculation we find that

$$\begin{array}{cc}\hfill m_0=& 95780971304118053647396689196894323976171195136475138\hfill \\ \hfill =& 2\times 3\times 11\times 43\times 251\times 281\times 1051\times 4051\times 86171\times 110251\hfill \\ & \times 347833278451\times 34010032331525251.\hfill \end{array}$$

By Lemma 1, we know that $m_0$ is practical. Let $x=2^{3^k}$, then we have $m_k=2(x^{175}+1)$ and $m_{k+1}=2(x^{525}+1)$. Since $k⩾0$, we have $x⩾2$. By Lemma 2, we have

$${\displaystyle \frac{x^{1050}+1}{x^{525}+1}}={\displaystyle \frac{x^{350}+1}{x^{175}+1}}{\Phi }_6\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{150}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{1050}\left(x\right).$$

(31)

Assume that $2(x^{175}+1)$ is practical, then we only need to show that

$$2(x^{175}+1){\Phi }_6\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{150}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{1050}\left(x\right)$$

is practical.

Note that

$$\begin{array}{cc}\hfill {\Phi }_6\left(x\right)=& x^2-x+1,\hfill \\ \hfill {\Phi }_{30}\left(x\right)=& x^8+x^7-x^5-x^4-x^3+x+1,\hfill \\ \hfill {\Phi }_{42}\left(x\right)=& x^{12}+x^{11}-x^9-x^8+x^6-x^4-x^3+x+1,\hfill \\ \hfill {\Phi }_{150}\left(x\right)=& x^{40}+x^{35}-x^{25}-x^{20}-x^{15}+x^5+1,\hfill \\ \hfill {\Phi }_{210}\left(x\right)=& x^{48}-x^{47}+x^{46}+x^{43}-x^{42}+2x^{41}-x^{40}\hfill \\ & +x^{39}+x^{36}-x^{35}+x^{34}-x^{33}+x^{32}-x^{31}\hfill \\ & -x^{28}-x^{26}-x^{24}-x^{22}-x^{20}-x^{17}+x^{16}\hfill \\ & -x^{15}+x^{14}-x^{13}+x^{12}+x^9-x^8\hfill \\ & +2x^7-x^6+x^5+x^2-x+1,\hfill \\ \hfill {\Phi }_{1050}\left(x\right)=& x^{240}-x^{235}+x^{230}+x^{215}-x^{210}+2x^{205}\hfill \\ & -x^{200}+x^{195}+x^{180}-x^{175}+x^{170}-x^{165}\hfill \\ & +x^{160}-x^{155}-x^{140}-x^{130}-x^{120}-x^{110}\hfill \\ & -x^{100}-x^{85}+x^{80}-x^{75}+x^{70}-x^{65}+x^{60}\hfill \\ & +x^{45}-x^{40}+2x^{35}-x^{30}+x^{25}+x^{10}-x^5+1.\hfill \end{array}$$

Since $x⩾2$, we have

$$\begin{array}{cc}\hfill x⩽& {\Phi }_6\left(x\right)⩽x^2,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill x^8⩽& {\Phi }_{30}\left(x\right)⩽2x^8,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill x^{12}⩽& {\Phi }_{42}\left(x\right)⩽2x^{12},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill x^{40}⩽& {\Phi }_{150}\left(x\right)⩽2x^{40},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{48}⩽& {\Phi }_{210}\left(x\right)⩽x^{48},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{240}⩽& {\Phi }_{1050}\left(x\right)⩽x^{240}.\hfill \end{array}$$

(37)

By (32)–(36), we have

$${\Phi }_6\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{150}\left(x\right){\Phi }_{210}\left(x\right)⩽8x^{110}⩽x^{113}⩽2(x^{175}+1)$$

(38)

and

$$2(x^{175}+1){\Phi }_6\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{150}\left(x\right){\Phi }_{210}\left(x\right)$$

is practical by Lemma 1.

By (32)–(36), we have

$$2(x^{175}+1){\Phi }_6\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{150}\left(x\right){\Phi }_{210}\left(x\right)⩾x^{284}.$$

(39)

By (37) and (39), we have

$${\Phi }_{1050}\left(x\right)⩽x^{240}⩽x^{284}⩽2(x^{175}+1){\Phi }_6\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{150}\left(x\right){\Phi }_{210}\left(x\right)$$

(40)

and

$$2(x^{175}+1){\Phi }_6\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{42}\left(x\right){\Phi }_{150}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{1050}\left(x\right)$$

is practical by Lemma 1.

Since $m_0$ is practical, we can show that $m_k$ is practical for every $k=0,1,2,\dots$ by an inductive process. Now the proof of sequence $2^{175\times 3^k+1}+2$ is completed.

2.4. Sequence 4

We shall prove that $2^{189\times 5^k+1}+2$ are practical for every $k⩾0$.

Write $m_k=2^{189\times 5^k+1}+2$ for $k=0,1,2,\dots$, then we only need to show that $m_k$ is practical for every $k=0,1,2,\dots$. By a computer calculation we find that

$$\begin{array}{cc}\hfill m_0=& 1569275433846670190958947355801916604025588861116008628226\hfill \\ \hfill =& 2\times 3^4\times 19\times 43\times 379\times 5419\times 87211\times 119827\times 77158673929\hfill \\ & \times 127391413339\times 56202143607667.\hfill \end{array}$$

By Lemma 1, we know that $m_0$ is practical. Let $x=2^{5^k}$, then we have $m_k=2(x^{189}+1)$ and $m_{k+1}=2(x^{945}+1)$. Since $k⩾0$, we have $x⩾2$. By Lemma 2, we have

$${\displaystyle \frac{x^{1890}+1}{x^{945}+1}}={\displaystyle \frac{x^{378}+1}{x^{189}+1}}{\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right){\Phi }_{630}\left(x\right){\Phi }_{1890}\left(x\right).$$

(41)

Assume that $2(x^{189}+1)$ is practical, then we only need to show that

$$2(x^{189}+1){\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right){\Phi }_{630}\left(x\right){\Phi }_{1890}\left(x\right)$$

is practical.

Note that

$$\begin{array}{cc}\hfill {\Phi }_{10}\left(x\right)=& x^4-x^3+x^2-x+1,\hfill \\ \hfill {\Phi }_{30}\left(x\right)=& x^8+x^7-x^5-x^4-x^3+x+1,\hfill \\ \hfill {\Phi }_{70}\left(x\right)=& x^{24}+x^{23}-x^{19}-x^{18}-x^{17}-x^{16}+x^{14}+x^{13}\hfill \\ & +x^{12}+x^{11}+x^{10}-x^8-x^7-x^6-x^5+x+1,\hfill \\ \hfill {\Phi }_{90}\left(x\right)=& x^{24}+x^{21}-x^{15}-x^{12}-x^9+x^3+1,\hfill \\ \hfill {\Phi }_{210}\left(x\right)=& x^{48}-x^{47}+x^{46}+x^{43}-x^{42}+2x^{41}-x^{40}\hfill \\ & +x^{39}+x^{36}-x^{35}+x^{34}-x^{33}+x^{32}-x^{31}\hfill \\ & -x^{28}-x^{26}-x^{24}-x^{22}-x^{20}-x^{17}+x^{16}\hfill \\ & -x^{15}+x^{14}-x^{13}+x^{12}+x^9-x^8\hfill \\ & +2x^7-x^6+x^5+x^2-x+1,\hfill \\ \hfill {\Phi }_{270}\left(x\right)=& x^{72}+x^{63}-x^{45}-x^{36}-x^{27}+x^9+1,\hfill \\ \hfill {\Phi }_{630}\left(x\right)=& x^{144}-x^{141}+x^{138}+x^{129}-x^{126}+2x^{123}-x^{120}\hfill \\ & +x^{117}+x^{108}-x^{105}+x^{102}-x^{99}+x^{96}-x^{93}-x^{84}\hfill \\ & -x^{78}-x^{72}-x^{66}-x^{60}-x^{51}+x^{48}-x^{45}+x^{42}\hfill \\ & -x^{39}+x^{36}+x^{27}-x^{24}+2x^{21}-x^{18}+x^{15}+x^6-x^3+1,\hfill \\ \hfill {\Phi }_{1890}\left(x\right)=& x^{432}-x^{423}+x^{414}+x^{387}-x^{378}+2x^{369}-x^{360}+x^{351}\hfill \\ & +x^{324}-x^{315}+x^{306}-x^{297}+x^{288}-x^{279}-x^{252}-x^{234}\hfill \\ & -x^{216}-x^{198}-x^{180}-x^{153}+x^{144}-x^{135}+x^{126}-x^{117}\hfill \\ & +x^{108}+x^{81}-x^{72}+2x^{63}-x^{54}+x^{45}+x^{18}-x^9+1.\hfill \end{array}$$

Since $x⩾2$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^4⩽& {\Phi }_{10}\left(x\right)⩽x^4,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill x^8⩽& {\Phi }_{30}\left(x\right)⩽2x^8,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill x^{24}⩽& {\Phi }_{70}\left(x\right)⩽2x^{24},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill x^{24}⩽& {\Phi }_{90}\left(x\right)⩽2x^{24},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{48}⩽& {\Phi }_{210}\left(x\right)⩽x^{48},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill x^{72}⩽& {\Phi }_{270}\left(x\right)⩽2x^{72},\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{144}⩽& {\Phi }_{630}\left(x\right)⩽x^{144},\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{x}^{432}⩽& {\Phi }_{1890}\left(x\right)⩽x^{432}.\hfill \end{array}$$

(49)

By (42)–(47), we have

$${\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right)⩽16x^{180}⩽x^{184}⩽2(x^{189}+1)$$

(50)

and

$$2(x^{189}+1){\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right)$$

is practical by Lemma 1.

By (42)–(47), we have

$$2(x^{189}+1){\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right)⩾{\displaystyle \frac{1}{2}}{x}^{369}⩾x^{368}.$$

(51)

By (48) and (51), we have

$${\Phi }_{630}\left(x\right)⩽x^{144}⩽x^{368}⩽2(x^{189}+1){\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right)$$

(52)

and

$$2(x^{189}+1){\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right){\Phi }_{630}\left(x\right)$$

is practical by Lemma 1.

By (42)–(48), we have

$$2(x^{189}+1){\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right){\Phi }_{630}\left(x\right)⩾{\displaystyle \frac{1}{4}}{x}^{513}⩾x^{511}.$$

(53)

By (49) and (53), we have

$${\Phi }_{1890}\left(x\right)⩽x^{432}⩽x^{511}⩽2(x^{189}+1){\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right){\Phi }_{630}\left(x\right)$$

(54)

and

$$2(x^{189}+1){\Phi }_{10}\left(x\right){\Phi }_{30}\left(x\right){\Phi }_{70}\left(x\right){\Phi }_{90}\left(x\right){\Phi }_{210}\left(x\right){\Phi }_{270}\left(x\right){\Phi }_{630}\left(x\right){\Phi }_{1890}\left(x\right)$$

is practical by Lemma 1.

Since $m_0$ is practical, we can show that $m_k$ is practical for every $k=0,1,2,\dots$ by an inductive process. Now the proof of sequence $2^{189\times 5^k+1}+2$ is completed.

Finally, we get Theorem 1 by combining the above four cases.

3. Proof of Theorems 1.3, 1.4 and 1.5

In order to prove Theorem 2, we need to show that at least one of the following five sequences generate infinitely many multiples of p for all prime $7⩽p<100$:

$$35\times 3^k+1,135\times 7^k+1,165\times 3^k+1,175\times 3^k+1,189\times 5^k+1.$$

By the following identities (where m is any positive integer)

$$\begin{array}{cc}\hfill 165\times 3^{5+6m}+1& \equiv 0(mod7),\hfill \\ \hfill 35\times 3^{3+10m}+1& \equiv 0(mod11),\hfill \\ \hfill 135\times 7^{3+12m}+1& \equiv 0(mod13),\hfill \\ \hfill 35\times 3^{8+16m}+1& \equiv 0(mod17),\hfill \\ \hfill 35\times 3^{17+18m}+1& \equiv 0(mod19),\hfill \\ \hfill 135\times 7^{20+22m}+1& \equiv 0(mod23),\hfill \\ \hfill 35\times 3^{24+28m}+1& \equiv 0(mod29),\hfill \\ \hfill 35\times 3^{27+30m}+1& \equiv 0(mod31),\hfill \\ \hfill 189\times 5^{32+36m}+1& \equiv 0(mod37),\hfill \\ \hfill 135\times 7^{7+40m}+1& \equiv 0(mod41),\hfill \\ \hfill 35\times 3^{3+42m}+1& \equiv 0(mod43),\hfill \\ \hfill 35\times 3^{11+46m}+1& \equiv 0(mod47),\hfill \\ \hfill 35\times 3^{1+52m}+1& \equiv 0(mod53),\hfill \\ \hfill 165\times 3^{21+58m}+1& \equiv 0(mod59),\hfill \\ \hfill 135\times 7^{50+60m}+1& \equiv 0(mod61),\hfill \\ \hfill 135\times 7^{33+66m}+1& \equiv 0(mod67),\hfill \\ \hfill 35\times 3^{11+70m}+1& \equiv 0(mod71),\hfill \\ \hfill 189\times 5^{57+72m}+1& \equiv 0(mod73),\hfill \\ \hfill 35\times 3^{2+78m}+1& \equiv 0(mod79),\hfill \\ \hfill 35\times 3^{24+82m}+1& \equiv 0(mod83),\hfill \\ \hfill 35\times 3^{69+88m}+1& \equiv 0(mod89),\hfill \\ \hfill 35\times 3^{40+96m}+1& \equiv 0(mod97),\hfill \end{array}$$

we complete the proof of Theorem 2. By combining those identities, we can also show that Conjecture Section 1 holds true for many other integers.

We can prove Theorem 3 by taking the sequences

$$4\times \left(2^{35\times 3^k+1}+2\right)=2^{35\times 3^k+3}+8$$

and

$$16\times \left(2^{35\times 3^k+1}+2\right)=2^{35\times 3^k+5}+32.$$

Similarly, for Theorem 4 we can take the sequence

$$2^{h-1}\times \left(2^{35\times 3^k+1}+2\right)=2^{35\times 3^k+h}+2^h$$

with a suitable h such that $35\times 3^k+h$ is a multiple of C.