On odd perfect numbers

R. Z. Ahmadullin

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2014), pp. 5-18

Résumé

A perfect number is a natural number equal to the sum of all its proper divisors (all positive divisors other than the number itself). Perfect numbers form a sequence: $6, 28, 496, 8128, 33550336, 8589869056, 137438691328~\dots$ Let $S=p_1^{a_1}\cdot p_2^{a_2}\cdots p_{n-1}^{a_{n-1}}\cdot p_n^{a_n}$ be a perfect number, where $p_i$ are primes, $a_i$ are some natural numbers, $a_i\geqslant 1$, $i=1, \dots, n$, and $n$ is the number of factors of the number $S$. Then \begin{equation} \frac{p_1^{1+a_1}-1}{(p_1-1)p_1^{a_1}}\cdot \frac{p_2^{1+a_2}-1}{(p_2-1)^*p_2^{a_2}}\cdots \frac{p_n^{1+a_n}-1}{(p_n-1)^*p_n^{a_n}}=2. \end{equation} Equation (1) is a Diophantine equation with an indefinite number of unknowns; it contains $2n$ unknowns, the value of $n$ (the number of factors of the number) is not fixed. This equation is equivalent to the two systems: \begin{equation} \left\{\begin{aligned} p_1=\frac1{\mathcal{Q}_1-1}\geqslant2, =\frac{-\ln(\mathcal{Q}_1-p_1(\mathcal{Q}_1-1))}{\ln(p_1)}\geqslant1;\\ \dots;\\ p_n=\frac1{\mathcal{Q}_n-1}\geqslant p(n), a_n=\frac{-\ln(\mathcal{Q}_n-p_n(\mathcal{Q}_n-1))}{\ln(p_n)}\geqslant1,\\ \end{aligned}\right. \end{equation} where $$ \begin{gathered} \mathcal{Q}_i=2\frac{(p_1-1)p_1^{a_1}}{p_1^{1+a_1}-1}\cdots \frac{(p_{i-1}-1)p_{i-1}^{a_{i-1}}}{p_{i-1}^{1+a_{i-1}}-1}\cdot \frac{(p_{i+1}-1)p_{i+1}^{a_{i+1}}}{p_{i+1}^{1+a_{i+1}}-1}\cdots \frac{(p_n-1)p_n^{a_n}}{p_n^{1+a_n}-1}=\\ =2\prod_{j=1}^{n\backslash i}\frac{(p_j-1)p_j^{a_j}}{p_j^{1+a_j}-1};\quad i=1,\dots,n, \end{gathered} $$ and \begin{equation} \left\{\begin{aligned} \frac{p_1^{1+a_1}-1}{(p_1-1)}=2^{\delta(a_1,p_1)}\prod_{j=1}^{n\backslash1}p_i^{a^{(1,j)}(a_1,p_1)};\dots;\\ \frac{p_n^{1+a_n}-1}{(p_n-1)}=2^{\delta(a_n,p_n)}\prod_{j=1}^{n-1}p_i^{a^{(n,j)}(a_n,p_n)};\quad \sum_{j=1}^n a^{(j)}=a_i; \quad i=1,\dots,n, \end{aligned} \right. \end{equation} where $\delta(a_1,p_1)$ is formally defined as follows: $$ \delta(a_1,p_1)= \begin{cases} 0, \text{ if } p_1=2,\\ 0, \text{ if } p_1\ne2 \text{ and } a_1\text{- even},\\ 1, \text{ if } p_1\ne2 \text{ and } a_1\text{- odd}. \end{cases} $$ With allowance for the fact that the factorization of natural numbers is determined uniquely, the system of equations (5) is a system of $2n$ equations and $2n$ unknowns (not with $(n^2+n)$ unknowns). The numbers $a^{(i,j)}$ are uniquely determined by a factorization function $F(p_1,a_1,i,j)$ and are considered as parameters. From the system of equations (2) we obtain the equation \begin{equation} a=-\frac{\ln\left(q-\frac1{q-1}(q-1)\right)^{-1}}{\ln\frac1{q-1}} \end{equation} at $2>q>1$. This function has an infinite number of (infinite) left discontinuities of the second kind at the points $q=(l+1)/1$ ($l\in\mathrm{N}$). Hypothetically, beginning from some values of $n$, most of exponents of $a_n$ in system (2) can be equal only to $1$. It is proved that for a given (fixed) value $n\geqslant3$ there exists only a finite number of odd perfect numbers.

Détail

Keywords: odd perfect number, amicable numbers, number theory.