Re: Find all solutions of a^4 - 4 a^2 b^2 + b^4 = square

In article <17802669.1150135764157.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Dale Shoults <the_shortest_possible_address@xxxxxxxx> wrote:

Find all solutions of a^4 - 4 a^2 b^2 + b^4 = square,
where a & b are coprime.

If (a,b) is a solution, so are (a,-b) and (b,a), so solutions come in
sets of 8 (except when a b = 0). Here is one representative of each octet:

(1, 2)
(4, 15)
(161, 442)
(22920, 50369)
(2706401, 21771082)
(19401465356, 37495194255)
(36867352480319, 241014273471602)
(2713767869349580560, 6160584655684657921)
(230411421854480432456639, 599328309772220125192082)
(54108065494048984622972655124, 224603401396311977137287538575)
(168116478779528111798480001300551839, 331803229851166469627898043962250282)
(54690892794338807630186727222541308756840, 1834835894556507914114598176948580439476929)
(19954369787472117896367095755263054539599807428319, 40576759762394305384329370667159957695834264439002)

etc. (The list is infinite, and yes, the visual pattern that you see about the
length of the digit strings does indeed continue forever.)

Keyword: "Elliptic Curves". (Rank 1, torsion Z/2Z x Z/2Z )

dave
.

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