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Determine (without checking all residues) whether the following congruence has a solution. x^2+ 1347x+ 5278≡0 (mod...

Determine (without checking all residues) whether the following congruence has a

solution. x^2+ 1347x+ 5278≡0 (mod 1452310211906532075867275721137)

Math   Advanced-Math   
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Answer #1

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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