Odd perfect numbers not divisible by $3$ are divisible by at least ten distinct primes

Odd perfect numbers have at least nine distinct prime factors

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

Non - Primes Are Recursively Divisible

In this article an inherent characteristics of the non-prime numbers of recursively holding the divisibility property is studied. The recursive property applies to any number system by virtue drawing some necessary conclusions. 2000 Mathematics Subject Classification: 11A51, 11K16.

Orientably regular maps with Euler characteristic divisible by few primes

Let G be a (2,m, n)-group and let x be the number of distinct primes dividing χ, the Euler characteristic of G. We prove, first, that, apart from a finite number of known exceptions, a nonabelian simple composition factor T of G is a finite group of Lie type with rank n ≤ x. This result is proved using new results connecting the prime graph of T to the integer x. We then study the particular ca...

Arithmetic Progressions with Common Difference Divisible by Small Primes

n(n + d) . . . (n + (k − 1)d) = by (1.1) in positive integers n, k ≥ 2, d > 1, b, y, l ≥ 3 with l prime, gcd(n, d) = 1 and P (b) ≤ k. We write d = D1D2 (1.2) where D1 is the maximal divisor of d such that all prime divisors of D1 are congruent to 1 ( mod l). Thus D1 and D2 are relatively prime positive integers such that D2 has no prime divisor which is congruent to 1 (mod l). Shorey [Sh88] pro...

Construction of Number Fields of Odd Degree with Class Numbers Divisible by Three, Five or by Seven