Non-congruent numbers with arbitrarily many prime factors congruent to $3$ modulo $8$

2-Selmer groups and the Birch–Swinnerton-Dyer Conjecture for the congruent number curves

“…• Lagrange: pqr is non-congruent when p, q ≡ 1 (mod 8), r ≡ 3 (mod 8), and ( p q ) = ( p r ) = −1 [4], • Iskra: p 1 · · · p is non-congruent when p j ≡ 3 (mod 8) for all j and ( p j p k ) = −1 for all j < k [8].…”

Families of non-congruent numbers with arbitrarily many prime factors

“…Thus, the sets N m and N m are distinct. A similar argument shows that for m 4 the integers in the sets N m are different from the integers in Iskra's theorem[4].2 Remark 1. The proof of Lemma 6 actually shows that T does not have full rank if m is odd and m 3.…”

“…Positive integers with a bounded number of prime factors have a density of zero as can be deduced from [7,Corollary], so it follows that there exist infinitely many odd squarefree non-congruent numbers with arbitrarily many prime factors. Of particular interest is the following family of non-congruent numbers due to Iskra [4], which contain arbitrarily many prime factors. Proposition 1.…”

“…This idea has already been investigated by Ono [9, Theorem 1], who for certain elliptic curves E gives a set of primes S of density 1/3 for which any twist of E by a product of primes in S produces a rank zero curve. Iskra's theorem on non-congruent numbers with arbitrarily many prime factors [4] is proved by using a complete 2-descent to show that the rank of the elliptic curves (1) is equal to zero. This requires the demonstration of the insolvability of arbitrarily many pairs of quadratic equations over Q, a method which involves a rapidly increasing amount of computation according to the number of prime factors of n and the values of the Legendre symbols relating those prime factors.…”

New series of odd non-congruent numbers

“…Many CN's and non-CN's have been determined (see refs. [1][2][3][4][5][6][7][8][9][10][11][12]), but most of them have at most 4 prime divisors p 1 , p 2 , . .…”