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I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$

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    I think writing $(x+iy)(x-iy) = (z-1)(z+1)$ might help.2011-10-22
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    You are not trying to solve Project Euler problem 224, are you? http://projecteuler.net/problem=2242011-10-23
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    @starblue. No, it came up when I tried to make up an example in hyperbolic geometry.2011-10-23
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    The question is essentially for which $z$ both $z-1$ and $z+1$ are the sum of two squares. There are some related questions asked before, such as http://math.stackexchange.com/questions/438818 and http://math.stackexchange.com/questions/46451.2014-09-08
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    I would look at Catalan’s [well-known] complete solution for $x_1^2+x_2^2+x_3^2=y_1^2$, and set $x_3 = a^2+b^2-c^2-d^2=1$. This implies another equal sums-of-squares equation $a^2+b^2=c^2+d^2+1$, so you probably have a nice orbit to chase. See also Spira's paper , _etc._.2014-10-06
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    @KierenMacMillan On my Blog, but the formula is. http://www.artofproblemsolving.com/blog/998042014-10-06

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