Determine (without checking all residues) whether the following congruence has a
solution. x^2+ 1347x+ 5278≡0 (mod 1452310211906532075867275721137)
Note that
So that if the equation has a solution then so must
That is, which it does
Also, divides the number so we must have
Checking we note that which is not a square modulo 431
Since the equation doesn't have a root modulo 431, it won't have a root modulo
Therefore, the equation has no integer solutions
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