If I take any 2 natural numbers: $a$ and $b$, such that $a \gt b$
How to solve this type of congruence equation to find $x$?: $$ a-x^2 \equiv 0\bmod{b} $$
Give me an example.
What properties of modular arithmetic can I use?
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Quadratic residue. – Simply Beautiful Art Mar 13 '20 at 02:49
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it's $x^2\equiv a\bmod b$ – J. W. Tanner Mar 13 '20 at 02:56
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So, basically you are looking for quadratic residue. There is no concrete way you can solve your equation for all such $a$ and $b$. But it is helpful to know the Legendre symbol when you deal with such type of equations.
I can provide one example of how you can approach (simple) problem: Solve equation $15-x^2 \equiv 0 \ (mod \ 7)$. Solution: $15-x^2 \equiv 0 \ (mod \ 3) \implies x^2 \equiv 15 \ (mod \ 7) \implies x^2 \equiv15-7\cdot2 \ (mod \ 7) \implies x^2 \equiv 1 \ (mod \ 7) \implies x^2-1 \equiv 0 \ (mod \ 7) \implies 3|(x-1)(x+1) \implies x \equiv\pm1 \ (mod \ 7)$.
You can post your exact problem, it will make it easier to understand.
Daniyar Aubekerov
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I suspected it could get complicated. But when you say "There is no concrete way you can solve your equation for all such a and b", did it mean that some of these equations can be solved? If so, can I post a new question with the specific problem? – Pedro Urday Mar 13 '20 at 03:02
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@PedroUrday Yes, it would be much better if you give us specific numbers so we can wrap up the properties. There is a way to point out how many $a$ and $b$ is possible, but it is just too broad to show which one in the residues are they. – Nikola Tolzsek Mar 13 '20 at 04:19