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**Nivla** Let $\displaystyle m_0 $ be the least possible $\displaystyle m $ for which there are $\displaystyle x$ and $\displaystyle y$ for which

$\displaystyle x^2 + y^2 = mp $, where p is a prime and $\displaystyle p \equiv 1 (mod4) $. Also $\displaystyle p \nmid x $, $\displaystyle p \nmid y $, $\displaystyle 0<m<p $ .

That there are such x,y and m is the subject of the previous exercise.

Thus, $\displaystyle x_0^2+y_0^2 = m_0p $ (1)

Show $\displaystyle m_0 \nmid x_0 $ and $\displaystyle m_0 \nmid y_0 $ unless $\displaystyle m_0 = 1 $.

By taking $\displaystyle m_0 $ as a factor of both $\displaystyle x_0 $ and $\displaystyle y_0 $, I find, from (1) above, that $\displaystyle p $ must be factored by $\displaystyle m_0 $. But p is prime. Therefore $\displaystyle m_0=1 $.

I feel I must have overlooked something, because I have not used the hypothesis that $\displaystyle m_0 $ is minimal. What have I overlooked, please?

The above problem is from GE Andrews's "Number Theory", ex. 2, Chapter 11-2.