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I just noticed something weird while messing around on Mathematica... For some pairs of primes $(p_1,p_2)$ the Diophantine equation $$ p_1 a^2 + p_2 b^2 = c^2 $$ has (non-trivial) solutions, whereas for other prime pairs it appears no solutions exist. Examples of prime pairs with solutions are $(2,7)$, $(3,13)$ and $(5,11)$; examples of prime pairs with no solutions (at least, for $c<1000$) are $(2,3)$, $(3,11)$ and $(5,13)$.

Assuming that there really are no solutions for the latter examples above, is there any known why of determining whether a given prime pair permits solutions?

Mark A
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  • There is a whole theory about quadratic forms, also including results on the numbers they represent. In your case, you are asking when the quadratic form $[p_1,p_2]$ over the integers represents a square (I assume you are looking for solutions $(a,b,c)$ and not fixing $c$ or anything?). I can't tell you just like this how to characterize the primes for which there are solutions, however, I am sure that at least partial results are known, somewhere in the theory of quadratic forms. – Dirk Apr 20 '17 at 08:17
  • Note that squares are always congruent to $0$, $1$ or $4$ modulo $8$. This can be used to prove that some of them have no solutions. I don't have a full answer, but if you want some more details on this, I can write a partial answer. – wythagoras Apr 20 '17 at 08:26
  • https://math.stackexchange.com/questions/1513733/solving-a-diophantine-equation-of-the-form-x2-ay2-byz-cz2-with-the-co/1514030#1514030 – individ Apr 20 '17 at 12:07

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