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Hello all!

I'm just wondering if there is an online encyclopedia of Diophantine equations (OEDE), analogous to the OEIS for sequences.

While trying to solve one Diophantine equation, I reduced the solution to that of a very similar Diophantine equation with smaller exponents (i.e., almost a descent, but not back to the exact same equation). What I want to do is type in the equation

$$a^2 + b^p = c^2$$

and find references, papers, solutions, etc. I know I can Google it, but there appears to be no standard format for equations, so the results are suspect, and what hits I do get are extremely time-consuming to obtain and filter through.

Thanks for any pointers! Kieren.

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Yes, Piezas has a collection of identities, mostly for infinite families of solutions. But he doesn't have much else like statements or proofs of non-existence of solutions. –  Seraj Jun 14 '13 at 18:02
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Don't know of anything online, but there is Mordell's book, Diophantine Equations, with an index of equations in the back. The book is about 50 years old, but still very useful. Someone should update it.... –  Gerry Myerson Jun 15 '13 at 5:54
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But will LaTeX Search find $a^3+b^3=c^3$ if you ask for $x^3+y^3=z^3$? –  Gerry Myerson Jun 19 '13 at 0:40
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A more up-to-date reference than Mordell's book would be Henri Cohen's 2 volume "Number Theory". Perusing through the contents pages you can find very many different Diophantine equations in the text. –  dke Jun 19 '13 at 13:07
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I'm reminded of this paper: arxiv.org/abs/1304.3866 –  Charles Jun 19 '13 at 19:26
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1 Answer

As mentioned in the comments, Piezas' site here has families of infinite solutions for many equations, and some complete characterizations.

There is a thread with a similar question here at math.stackexchange which may be helpful, if you remove the "online" component of your question. The answers in the thread state:

Diophantine Equations by Mordell

The Algorithmic Resolution of Diophantine Equations by Smart

Algorithms For Diophantine Equations by de Weger

History of the Theory of Numbers, Volume 2 by Dickson

EDIT: Another of the replies said Number Theory, Volume 1 and 2 by Cohen, but that he had not read either. Since someone just added it to the comments in this thread, I'm now including it here for completeness.

I would add that An Introduction to Diophantine Equations by Andreescu, designed for math olympiad training, has many equations and their solutions.

One of the answers in the math.stackexchange thread says "I imagine a complete catalog and bibliography today, bang up to date but in the same detail as Dickson, would require dozens of volumes and would be more a WIkipedia style project than anything one could reasonably contemplate publishing in book form."

This resonates with your online idea, so perhaps someone should initiate it. Certainly it would be very useful, especially to prevent wasting research time on rediscoveries.

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