For square-free n = p1...pt q with pi ≡1 mod 8 and q≡3 mod 8, if n is a congruent number then the 2-part of the class number of Q(sqrt(-n)) is congruent modulo 2^k to that of Q(sqrt(-p1...pt)) via a modified Rédei matrix.
Genocchi , Note analitiche sopra tre scritti inediti di leonardo pisano pubblicati da baldassarre boncompagni , Annali di Scienze Matematiche e Fisiche, 6 (1855), pp
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A necessary condition for a congruent number of the form $8k+3$
For square-free n = p1...pt q with pi ≡1 mod 8 and q≡3 mod 8, if n is a congruent number then the 2-part of the class number of Q(sqrt(-n)) is congruent modulo 2^k to that of Q(sqrt(-p1...pt)) via a modified Rédei matrix.