Some families of non-congruent numbers

Some Families of Non-congruent Numbers

In this article we study the Tate-Shafarevich groups corresponding to 2-isogenies of the curve Ek : y 2 = x(x2 − k2) and construct infinitely many examples where these groups have odd 2-rank. Our main result is that among the curves Ek, where k = pl ≡ 1 mod 8 for primes p and l, the curves with rank 0 have density ≥ 1 2 .

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...

Average Kissing Numbers for Non-congruent Sphere Packings

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 ...

Congruent subsets of infinite sets of natural numbers

This improves an earlier result of the first author for k = n = 2, namely Theorem II below, which dealt with B2 -sequences. This notion arose from a paper of Sidon [6]. A strictly increasing sequence A of natural numbers a,, a2 . . . . is called a B2-sequence, if for any two pairs (i, j) + (k, m) of natural numbers i <j, k < m there holds aj a i + ak a„„ in short, if A has no double-differences...