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Families of non-congruent numbers with arbitrarily many pairs of prime factors. (English) Zbl 1459.11085

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.

MSC:

11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
14G05 Rational points
14H25 Arithmetic ground fields for curves

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