On odd perfect numbers

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 ... Let S = pi • p" - р"- • pL" be a perfect number, where p _t are primes, a _i are some natural numbers, a,> 1, i = 1,...,n, and n is the number of factors of the number S. Then р2+ " -1 1+" l+i" I P _n -1 -1 Pi (1) - = 2. (Pl - 1K (p ₂ - 1)* p" (Pn - 1)* p" 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: 'p^-L- > 2, l = i Pi -1)) > l; ^ ln( Pi Qi -1 1 QL i > P (n), (2) -l"(Qn - Pn (Qn - 1)) > (Pn ' (Pn - ') Pi" Pn "" - 1 (-i). _г=^ where _Q = ₂ (Pi - i^Pi" (Pi-1 - ^рГ-- (+1 - ^nOl ^ -1 pi^i-i -1 p^+i -1 1 n\i = П 1+a j -j=i Pj 1 and n\1 (1 j) 2 ( l Pi ^Пр" ОьPi . j=i (3) (j). Z" j=1 a,; i = 1, ( Pi -1) 1+a n = 9 0» Pn _n" (Pn - 1)= {=Г 1+ " 1 Pi where 5 (a ₁, р ₁) is formally defined as follows: Г 0, if р ₁ = 2, S(a ₁, р ₁ ) = ^0, if р ₁ * 2 and a ₁ - even, [ 1, if р ₁ * 2 and a ₁ - odd. 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 + n) unknowns). The numbers a are uniquely determined by a factorization function F (р ₁, a ₁, i, j) and are considered as parameters. From the system of equations (2) we obtain the equation inf q ^(q ^ a = - q 1- - (4) ln- q -1 at 2 > q > 1. This function has an infinite number of (infinite) left discontinuities of the second kind at the points q = (l + 1) / l (le 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 > 3 there exists only a finite number of odd perfect numbers.

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