TITLE: A curious pattern on primes congruent to $1$ mod $4$?
QUESTION [8 upvotes]: 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 ?

REPLY [5 votes]: Let $a+b=35$ and $a-b=9$. Neither is prime.
Then $a=22$ and $b=13$, and the sum $(22)^2+(13)^2$ is the prime $653$.
Remark: For nice examples of apparent patterns that disappear when we look at larger numbers, please see Richard Guy's The Strong Law of Small numbers.