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Do there exist integers $m$ and $n$ that satisfy the equation $m^{2}+1954=n^{2} ?$

#
Suppose that $(n-m)(n+m)=n^{2}-m^{2}=1954$. The numbers $n-m$ and $n+m$ are of the same parity, so their product is either odd or divisible by 4. But 1954 is even and not divisible by 4.

## Answer

They do not exist.