Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?
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asked Nov 17, 2015 at 10:03
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$\begingroup$ Could you tell us a little more why you are interested in this question, please? $\endgroup$– Todd TrimbleCommented Nov 17, 2015 at 13:10
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$\begingroup$ I have read several descriptions of the link but none of them give any historical background. I did a course on History of Science as an undergraduate, so have always been interested in this aspect of Maths. $\endgroup$– Allan MacLeodCommented Nov 18, 2015 at 7:30
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$\begingroup$ @AllanMacLeod Perhaps History of Science and Mathematics would be a good home for this question? $\endgroup$– DanuCommented Nov 18, 2015 at 18:59
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$\begingroup$ @AllanMacLeod: Can you kindly look at this MSE post on the related concordant forms/numbers? $\endgroup$– Tito Piezas IIICommented Dec 21, 2015 at 14:47
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2 Answers
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I guess the first to make the explicit connection between $n$ being a congruent number and the existence of a nontrivial rational point on the ellliptic curve $y^2 = x^4 - n^2$ was Edouard Lucas in Recherches sur plusieurs ouvrages de Léonard de Pise et sur diverses questions d'arithmétique supérieure in 1877. But this was before the group law on elliptic curves was discovered by Juel and others at the end of the 19th century. The first who really exploited this connection was Kurt Heegner in his famous article Diophantische Analysis und Modulfunktionen (1952) in which he (essentially) solved the class number 1 problem for complex quadratic number fields. Heegner also used the Weierstrass form of elliptic curves.
answered Nov 26, 2015 at 18:33
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According to this mathematically historical paper the origin of this observation goes to Fermat and his method of infinite descent, though Mordell (as noted by Weil) was the first to make it explicit in this particular form.