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The background of this question is this: Fermat proved that the equation, $$x^4+y^4 = z^2$$

has no solution in the positive integers. If we consider the near-miss, $$x^4+y^4-1 = z^2$$

then this has plenty (in fact, an infinity, as it can be solved by a Pell equation). But J. Cullen, by exhaustive search, found that the other near-miss, $$x^4+y^4+1 = z^2$$

has none with $0 < x,y < 10^6$.

Does the third equation really have none at all, or are the solutions just enormous?

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    I assume you're talking about this: http://members.bex.net/jtcullen515/Math10.htm? I would guess that the reason for the computer search is that it's an open problem that nobody knows the answer to.2011-01-09
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    From the [faq](http://math.stackexchange.com/faq): Please don't use signatures or taglines in your posts. Every post you make is already "signed" with your standard user card, which links directly back to your user page. Your user page belongs to you — fill it with interesting information about your interests, links to cool stuff you've worked on, or whatever else you like!2011-01-09
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    Yes, Hans, that is his site. Eq.2 can be solved via a Pell equation, so it has an infinity of positive integer solutions. Cullen and I tried to find parametrizations for p^4+q^2+1 = r^2, in the hope that we could specialize "q", but the easy identity I found had a "q" that was never a square.2011-01-09
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    I haven't looked into this at all, but does the way Fermat proved there are no solutions to x^4+y^4=z^2 help for x^4+y^4=z^2-1=(z+1)(z-1)?2011-01-24
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    mod 5 reduction could be a good starting point (since $\phi(5)=4$). Unless I missed something, $5|x, y$ and $z=5^{4k}m \pm 1$, I wonder if this helps.2011-04-09
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    @user9176: this is easily extended to conclude $10|x,y$ and that $z \equiv \pm 1 \text { or } \pm 1249 \mod 5000$.2011-04-15
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    @Tito The link provided by Hans Lundmark actually show that there are no solutions for $0 < x, y < 10^7$.2011-04-15
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    Relevant: http://mathoverflow.net/q/61794/12357 and http://meta.mathoverflow.net/q/729/123572013-08-28

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