ON ODD PERFECT NUMBERS

It is shown here that if n is odd and perfect, then n has a prime divisor which exceeds 11200.1. Introduction.If a(ri) denotes the sum of the positive divisors of the natural number n, then n is said to be perfect ifTwenty-four even perfect numbers are known at present, but whether or not an odd perfect number exists is an open question.However, many necessary conditions for the existence of an odd perfect number are known.The oldest of these is due to Euler who proved that if n is odd and perfect, thenwhere the/?, are distinct primes, /?" = a" = I (mod 4), and a, is even for i > 0. In 1944 Kanold [2] showed that at least one of the /?, is greater than or equal to 61.Our purpose here is to show that Kanold's result can be significantly improved.Thus, we shall prove the following result.Theorem.Ifn is an odd perfect number, then n is divisible by aprime which exceeds 11200.Our proof is computational in nature in as much as it is based on a search for the prime factors less than 11200 of <r(pa) for a large number of values of/? and a.This search was carried out using the CDC 6400 at the Temple University Computing Center and utilized the fact that cr(/?) = J\ F~P) where Fd is the d\h cyclotomic polynomial and d runs over the divisors of a + 1 which exceed 1.The theorems concerning the prime divisors of cyclotomic polynomials which may be found in Section 48 of [3] facilitate the search. Some Preliminaries.According to a theorem of Kanold [1], if P is the greatest of the/?, in (2) and is the maximum of the a,, then P > 2(/3 + 1).It follows that for our purpose we may assume that each a, is less than 5599.Thus, in the remainder of this paper, n will always denote an odd perfect number whose canonical factorization is given by ( 2), with every a, < 5599.If p" | n but pa +' X n, we shall write p" \\ n.In this case, we note from (1) and the fact that the <r-function is multiplicative that every odd prime which divides o-(p") also divides n.3. Phase I. Suppose that/? is a prime and a is a natural number.We shall say that /?" is (5599; 11200)-acceptable or simply acceptable if a < 5599 and every prime

A 1913 theorem of Dickson asserts that for each fixed natural number k, there are only finitely many odd perfect numbers N with at most k distinct prime factors. We show that the number of such N is bounded by 4k2.

An odd perfect number, N N , is shown to have at least nine distinct prime factors. If 3 ∤ N 3\nmid N then N N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.

One of the oldest unsolved mathematical problems is the following one: Are there odd perfect numbers? So many interesting necessary conditions for an odd integer to be perfect have been found out. A bibliography of previous work is given by McCarthy [5]. Throughout this paper n denotes an odd perfect number. The following results have been proved in [1 ] and [2 ] respectively: (i) 1/2 < EvIn (1/p) <2 log (r/2) (-903), (ii) n must be of the form 12t+1 or 36t+9. The bounds for E..n (l/p) given in [1 ] have been improved in [3 ] as

Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding 10 7 10^7 . Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding 10 8 10^8 .

On the multiplicative structure of odd perfect numbers

The Descartes-Frenicle-Sorli conjecture predicts that = 1 if 2 is an odd perfect number with Euler prime . In this note, we present some further conditions equivalent to this conjecture.

Consider the computer: It can be used to develop a new method of personal identification based on dental bitemarks that may aid efforts in law enforcement and forensic dentistry; it can be designed so that it runs multiple interacting programs simultaneously almost three times faster than a single microprocessor-based system; and when combined with theoretical number formulas, it can show that an odd perfect number - in which the sum of its factors is twice the original number-must be at least 1079.

We make a table of odd integers N with five distinct prime factors for which 2 − 10 − 12 &gt; σ ( N ) / N &gt; 2 + 10 − 12 2 - {10^{ - 12}} &gt; \sigma (N)/N &gt; 2 + {10^{ - 12}} , and show that for such N | σ ( N ) / N − 2 | &gt; 10 − 14 N\;|\sigma (N)/N - 2| &gt; {10^{ - 14}} . Using this inequality, we prove that there are no odd perfect numbers, no quasiperfect numbers and no odd almost perfect numbers with five distinct prime factors. We also make a table of odd primitive abundant numbers N with five distinct prime factors for which 2 &gt; σ ( N ) / N &gt; 2 + 2 / 10 10 2 &gt; \sigma (N)/N &gt; 2 + 2/{10^{10}} .

Let o{n) denote the sum of the positive divisors of the integer n and s(n) the sum of its aliquot parts, that is, of the divisors of n other than n itself; then, s(n) =■■ o{ri) n. Greek mathematicians had considered the problem of the determination of a perfect number (teXeloc dpt^oc), that is, a number n such that s(n) = n. Euclid demonstrated that if 2m 1 is a prime, then 2"I~I(2m 1) is perfect. Euler proved that any even perfect number must have this form, so that the question of the generality of Euclid's formula depends upon the existence of odd perfect numbers. This remains an open question today since no odd perfect number has been found but no demonstration of their non-existence has been found either. The idea of considering these perfect numbers goes back to the Pythagoreans. Such is also the case for the so-called amicable numbers (ciXoi apiifyoi), that is, pairs of numbers at,, n2 such that each is the sum of the aliquot parts of the other, i.e., s(ni) = n2 and s(n2) = nA or (t(az,) v{n2) =:z nx + n2. The Pythagoreans apparently only knew of the pair 220,284, and a method for finding further pairs (actually having the very restricted form nx = 2hp, n2 2hqr, with /?, q, r different primes) was not discovered before the ninth century, by Thabit ibn Qurra. His rule was rediscovered in seventeenth century Europe, from which time research went further. Thus, Euler determined some sixty pairs, but not subject to the former restriction on their prime factorization. Two further questions in relation to the sums of divisors arose during Islamic times:

It is proved here that every odd perfect number has a prime factor greater than 100110.

An infinite tree-generating " q -algorithm" is defined, which if executed would enumerate all odd perfect numbers (opn’s). A truncated execution shows that any opn has either some component p a &gt; 10 18 {p^a} &gt; {10^{18}} , with a even, or no divisor &gt; 7 &gt; 7 ; hence any opn must be &gt; 10 36 &gt; {10^{36}} .

We describe an algorithm for proving that there is no odd perfect number less than a given bound K (or finding such a number if one exists). A program implementing the algorithm has been run successfully with K = 10 160 K = {10^{160}} , with an elliptic curve method used for the vast number of factorizations required.

Every Odd Perfect Number Not Divisible by 3 has at Least Eleven Distinct Prime Factors.

Odd perfect numbers and their Euler prime