Families of Non-θ-congruent Numbers with Arbitrarily 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 ...

Colloquia Patavina on the Twin Prime Conjecture by János Pintz

The twin prime conjecture is one of the oldest (if not the oldest) and most famous unsolved problem of the whole mathematics. We call a pair of primes a twin prime pair if their difference is exactly 2, that is, if they are consecutive odd numbers, which are both prime numbers. The twin prime conjecture asserts that there are infinitely many twin primes, that is, we can find above arbitrary lar...

New Constructions of Self-Complementary Cayley Graphs

Self-complementary Cayley graphs are useful in the study of Ramsey numbers, but they are relatively very rare and hard to construct. In this paper, we construct several families of new self-complementary Cayley graphs of order p4 where p is a prime and congruent to 1 modulo 8.

On the Value Set of n! Modulo a Prime

We show that for infinitely many prime numbers p there are at least log log p/ log log log p distinct residue classes modulo p that are not congruent to n! for any integer n.

Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes