Observation about twin primes: is it true? If so, why?

Question

I noticed today that every set of twin primes except for $(3,5)$ and $(5,7)$ seems to have one of the two primes that can be represented by the sum of two squares. For example: \begin{eqnarray*} 13=3^2+2^2 \\ 17=4^2+1^2 \\ 29=5^2+2^2 \end{eqnarray*}

... (sum of two squares works for all of these, I checked) \begin{eqnarray*} 281&=&16^2+5^2 \\ 313 &=&13^2+12^2 \\ 349&=&18^2+5^2 \end{eqnarray*}

I am wondering if this works for all twin primes. If so, can someone explain why (because I have thought about it some but nothing really comes to mind)? Thank you.

Answer 1

HINT: Consider the following theorem:

An odd prime $p$ can be represented as a sum of $2$ squares if and only if it is congruent to $1$ modulo $4$.

Now notice that all odd primes must be congruent to either $1$ or $3$ modulo $4$, and if $p$ is congruent to $3$ mod $4$, then $p+2$ is congruent to $1$ mod $4$.

Your conjecture is true, and it follows from the above. Can you see why?

Answer 2

You can generalize this result to the following:

Every prime pair $(p,p+k)$ which k is congruent to $2$ mod $4$ satisfies to your claim i.e. one of two primes $p$ , $p+k$ can be represented by the sum of two squares.