Is there some indicator for false positives on the Legendre symbol?For example, ( 35 13 ) = 1yet x 2 − 35 y 2 = 13 has no solutions in integers. For a non-UFD like this, is there some way to take the Legendre symbol, add to it or multiply it by something else, like maybe something to the power of the class number to get the − 1 meaning that 13 is irreducible in this particular domain of algebraic integers?

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Is there some indicator for false positives on the Legendre symbol?For example,      (            35      13        )    =  1yet       x    2    −  35      y    2    =  13 has no solutions in integers. For a non-UFD like this, is there some way to take the Legendre symbol, add to it or multiply it by something else, like maybe something to the power of the class number to get the   −  1 meaning that   13 is irreducible in this particular domain of algebraic integers?

Salma Bradley · Accepted Answer

At the very least, if       x    2    −  d      y    2    =  n has a solution for some d,n square-free, then if   p  ∣  d and   q  ∣  n are prime then   d is a square modulo   q and   n is a square modulo   p. Your example fails this test, since   13 is not a square modulo neither   5 nor   7.But this is known to not be enough. Even the case   n  =  −  1 is very tricky, and has proved somewhat intractable over the years, except by computing the continued fraction for       d   explicitly.

Is there some indicator for false positives on the Legendre symbol? For example, (

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