Required condition for a congruent number: \(pq\) with primes \(p \equiv 1 \pmod 8\) and \(q \equiv 3 \pmod 8\) (Q6585659)

In the paper under review, the author gives a necessary condition for an integer \(n\) of the form \(n=pq\) with \((p,q)\equiv (1,3)\pmod 8\) to be a \textit{congruent number}, i.e., a positive integer that can be realized as the area of a right triangle with rational side lengths. Let \((p,q)\) be as above, and denote the class number of \({\mathbb Q}(\sqrt{-2pq})\) by \(h(-2pq).\) Then the author states the primary findings as follows:\N\NTheorem 1.1. Assume that \(n=pq,\) where both \(p\) and \(q\) are primes such that \((p,q)\equiv (1,3)\pmod 8.\) If \(n\) is a congruent number, then one has \N\[\Nh(-2pq)\equiv p-1\pmod{16}.\N\]\NLet \(E_n\) be the congruent number elliptic curve. One knows that \(n\) is a congruent number if \(E_n\) has positive Mordell-Weil rank \(r(n).\) To prove the above result, the author adapts the complete \(2\)-descent method [\textit{P. Serf}, Proc. Colloq. Debrecen/Hung. 1989, 227--238 (1991; Zbl 0736.11017)] into a more convenient form that involves solving Diophantine equations with integer solutions (see Lemmas 2.4-2.7), where in Lemma 2.7, the author has used a result of \textit{J. Lagrange} [Theorie des Nombres, Fasc. 1, Expose 16, 17 p. (1975; Zbl 0328.10013)] stating that for \((p,q)\) as above, if \(\left(\frac q p\right)=-1,\) then \(pq\) is not a congruent number. As a link to class number, the author uses the following result [\textit{P. Kaplan}, J. Reine Angew. Math. 283/284, 313--363 (1976; Zbl 0337.12003)].\N\NProposition 3.1. Suppose \(p,q\) are primes of the form \((p,q)\equiv (1,3)\pmod 8\) and \(\left(\frac q p\right)=1.\) Then one has \N\[\Nh(-2pq)\equiv 4\left(1-\left(\frac{2q}{p}\right)_4\right)\pmod{16}.\N\]\NUsing a result of \textit{C. F. Gauss} [Disquisitiones arithmeticae. (Translated by Arthur A. Clarke). New Haven-London: Yale University Press (1966; Zbl 0136.32301)] for representing primes of the form \(8k+1\) and manipulating with quadratic and quartic reciprocity, the author establishes the following lemma which together with Proposition 3.1 proves Theorem 1.1.\N\NLemma 3.3. Suppose \(p,q\) are primes satisfying \((p,q)\equiv (1,3)\pmod 8\) and \(E_{pq}({\mathbb Q})\) has a point of infinite order. Then \N\[\N4\left(1-\left(\frac{2q}{p}\right)_4\right)\equiv p-1\pmod{16}.\N\]\NIn Section 4, the author gives evidence to the prediction regarding \(r(n)=2\) of the associated elliptic curve \(E_n\) in the situation considered in the paper, namely the author lists computational results displaying \(r(n)=2\) and \(h(-2pq)\) for \(n=pq\) as in Theorem 1.1 with \(n\leq 28907.\)\N\NRemark. The condition given in Theorem 1.1 is necessary but not sufficient. As an example, for \(n=12931=pq\) with \(p=193,q=67\) and \(\left(\frac q p\right)=1,\) one has \(h(-2pq)=96\equiv p-1\pmod{16},\) but \(n\) is not a congruent number.