What does this research mean for the field?

If primes p and q satisfy specific congruences, then the number 2pq is not a congruent number under certain conditions related to the class number of imaginary quadratic fields. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to determine conditions under which the product 2pq does not yield a congruent number for specific prime pairs.

February 25, 2026Open Access

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NONCONGRUENT NUMBERS OF THE FORM 2pq WITH pq 5\ (mod 8)

GJGuifen JieQingdao University of Science and Technology

Key Points

The aim is to determine conditions under which the product 2pq does not yield a congruent number for specific prime pairs.
Analyzed conditions on primes p and q modulo 8
Used results from Lagrange and Qin on congruent numbers
Examined class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$
Demonstrated that for $(p,q) \equiv (1,5) \pmod 8$ and $(\frac{q}{p})=1$, 2pq is not a congruent number
Showed that for $(p,q) \equiv (7,3) \pmod 8$ and $(\frac{q}{p})=1$, 2pq is not a congruent number under specified conditions

Abstract

Abstract Let p and q be two primes with (p, q) (1, 5) or (7, 3) 8. Lagrange ‘Nombres congruents et courbes elliptiques’, Séminaire Delange-Pisot-Poitou. Théorie des Nombres 16 (1) (1974–1975), Article no. 16 and Qin ‘Congruent numbers, quadratic forms and K₂ ’, Math. Ann. 383 (3–4) (2022), 1647–1686 showed that if (qp) =-1, then 2pq is not a congruent number. By using Qin’s method, we prove that if (p, q) (1, 5) 8 and (qp) =1 with h (-pq) p-1 16, then 2pq is not a congruent number; if (p, q) (7, 3) 8 and (qp) =1 with h (-2pq) p+1 16, then 2pq is not a congruent number. Here, h (-d) denotes the class number of the imaginary quadratic field Q (-d).

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NONCONGRUENT NUMBERS OF THE FORM 2pq WITH pq 5\ (mod 8)

Key Points

Abstract

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