Das, Shamik; Saikia, Anupam Families of non-congruent numbers with arbitrarily many pairs of prime factors. (English) Zbl 1459.11085 Integers 20, Paper A55, 12 p. (2020). A natural number \(n\) is called a congruent number if it is the area of a right triangle with rational lengths. Recall that \(n\) is a non-congruent number if the elliptic curve \(E_n : y^2 = x^3- n^2x\) has Mordell-Weil rank 0. Let \(p_1, \ldots,p_t\) and \(q_1, \ldots, q_t\) be distinct primes such that all pairs \((p_j, q_j)\) are equivalent either to \((1, 3)\) or to \((5, 7)\) modulo 8. Suppose that \((q_j/q_i) = -1\) if \(i>j\), \((p_i/p_j) = 1\), if \(i\neq j\), and \((p_i/q_j) = 1\), if \(i\neq j\) and \(-1\), otherwise. Then it is proved that \(n = (p_1q_1)\cdots (p_tq_t)\) is a non-congruent number.Further, let \(H_t\) be the collection of positive integers with prime factorization \((p_1q_1)\cdots(p_tq_t)\), where all the pairs \((p_j, q_j)\) are equivalent to \((1, 3)\) modulo 8 and satisfy the above conditions. Then, it is proved that for any natural number \(t\), the set \(H_t\) contains infinitely many elements. The analogous statement for pairs \((p_j, q_j)\) equivalent to \((5, 7)\) modulo 8 holds as well. Some examples of non-congruent numbers are provided.For the proof of these results, it is shown how 2-descent leads to an “unsolvability condition” for ruling out existence of non-torsion rational points on \(E_n\), and so the unsolvability condition holds for composite numbers of the form given above. Reviewer: Dimitros Poulakis (Thessaloniki) Cited in 1 ReviewCited in 1 Document MSC: 11D25 Cubic and quartic Diophantine equations 11G05 Elliptic curves over global fields 14G05 Rational points 14H25 Arithmetic ground fields for curves Keywords:congruent number; Mordell-Weil rank; elliptic curve Software:Magma PDFBibTeX XMLCite \textit{S. Das} and \textit{A. Saikia}, Integers 20, Paper A55, 12 p. (2020; Zbl 1459.11085) Full Text: Link References: [1] hence n is non-congruent. We further verify that (5 • 7) • (29 • 79), (5 • 7) • (821 • 151) and (29 • 79) • (821 • 151) are non-congruent too, as implied by Theorem 1. [2] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3-4) (1997), 235-265. · Zbl 0898.68039 [3] B. Iskra, Non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 168-169. · Zbl 0877.11014 [4] J. Lagrange, Nombres congruents et courbes elliptiques, in Séminaire Delange-Pisot-Poitou (16e année: 1974/75), Théorie des Nombres, Fasc. 1, 16, 1975. · Zbl 0328.10013 [5] P. Monsky, Mock Heegner points and congruent numbers, Math. Z. 204(1) (1990), 45-67. · Zbl 0705.14023 [6] L. Reinholz, B. K. Spearman, and Q. Yang, Families of even non-congruent numbers with prime factors in each odd congruence class modulo eight, Int. J. Number Theory 14(3) (2018), 669-692. · Zbl 1429.11108 [7] P. Serf, Congruent numbers and elliptic curves, in Computational Number Theory (Debrecen, 1989), de Gruyter, Berlin, 1991. · Zbl 0736.11017 [8] J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer, Dordrecht, 2009. · Zbl 1194.11005 [9] Y. Tian, Congruent numbers and Heegner points, Camb. J. Math. 2(1) (2014), 117-161. · Zbl 1303.11067 [10] J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math. 72(2) 1983, 323-334. · Zbl 0515.10013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.