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**demode** I need some help with the following problem:

**(a)** Prove that the equation $\displaystyle x^2 + 23y^2 = 41$ does not have solutions in integers.

**(b)** If (1/3, 4/3) and (9/4, 5/4) are two rational solutions of this equation, find a solution to this equation in $\displaystyle \mathbb{Z}_{568}$ and another one in $\displaystyle \mathbb{Z}_{657}$.

I have not encountered this type of problems before, so I'm unsure how to get started on either parts. Any help to get me started on them is greatly appreciated.