<s>[INST] A curious pattern on primes congruent to $1$ mod $4$? It is known that every prime $p$ that satisfies the title congruence can be expressed in the form $a^{2} + b^{2}$ for some integers $a,b$, and unique factorisation in $Z[i]$ ensures exactly one such representation for each $p \equiv 1 \mod 4$.
It seems at least one of $a-b, a+b$ is always a prime ? Is there any mathematical explanation for this ?
 [/INST] This is not true for $p=41$ because, except for permutations and signs, the only possible values are $a=5$ and $b=4$ but $a-b=1$ and $a+b=9$, not primes.
Another counterexample is $p=353$, for which $a=17$ and $b=8$ but $a-b=9$ and $a+b=25$, both squares!
Here are the first few counterexamples:
\begin{array}{rccccc}
p & a & b & a-b & a+b \\
41 & 5 & 4 & 1 & 9 \\
113 & 8 & 7 & 1 & 15 \\
313 & 13 & 12 & 1 & 25 \\
353 & 17 & 8 & 9 & 25 \\
613 & 18 & 17 & 1 & 35 \\
653 & 22 & 13 & 9 & 35 \\
677 & 26 & 1 & 25 & 27 \\
761 & 20 & 19 & 1 & 39 \\
857 & 29 & 4 & 25 & 33 \\
977 & 31 & 4 & 27 & 35 \\
\end{array}
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