Primitive abundant and weird numbers with many prime factors

Congruent numbers with many prime factors.

Mohammed Ben Alhocain, in an Arab manuscript of the 10th century, stated that the principal object of the theory of rational right triangles is to find a square that when increased or diminished by a certain number, m becomes a square [Dickson LE (1971) History of the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In modern language, this object is to find a rational point of infinite ...

On the distribution of primitive abundant numbers

Least primitive root of prime numbers

Let p be a prime number. Fermat's little theorem [1] states that a^(p-1) mod p=1 (a hat (^) denotes exponentiation) for all integers a between 1 and p-1. A primitive root [1] of p is a number r such that any integer a between 1 and p-1 can be expressed by a=r^k mod p, with k a nonnegative integer smaller that p-1. If p is an odd prime number then r is a primitive root of p if and only if r^((p-...

Carmichael Numbers With Three Prime Factors

A Carmichael number (or absolute pseudo-prime) is a composite positive integer n such that n|an − a for every integer a. It is not difficult to prove that such an integer must be square-free, with at least 3 prime factors. Moreover if the numbers p = 6m + 1, q = 12m + 1 and r = 18m + 1 are all prime, then n = pqr will be a Carmichael number. However it is not currently known whether there are i...

Families of Non-θ-congruent Numbers with Arbitrarily Many Prime Factors

The concept of θ-congruent numbers was introduced by Fujiwara [Fu97], who showed that for primes p ≡ 5, 7, 19 (mod 24), p is not a π/3-congruent number. In this paper we show the existence of two infinite families of composite non-π/3-congruent numbers and non-2π/3-congruent numbers, obtained from products of primes which are congruent to 5 modulo 24 and to 13 modulo 24 respectively. This is ac...