Definition:Pythagorean Prime

Definition

A Pythagorean prime is a prime number of the form:

$p = 4 n + 1$

where $n \in \Z_{\ge 0}$ is a positive integer.

Sequence

The sequence of Pythagorean primes begins:

$\begin{array} {r | r | r} p & 4 n + 1 & a^2 + b^2 \\ \hline 5 & 4 \times 1 + 1 & 2^2 + 1^2 \\ 13 & 4 \times 3 + 1 & 3^2 + 2^2 \\ 17 & 4 \times 4 + 1 & 4^2 + 1^2 \\ 29 & 4 \times 7 + 1 & 5^2 + 2^2 \\ 37 & 4 \times 9 + 1 & 6^2 + 1^2 \\ 41 & 4 \times 10 + 1 & 5^2 + 4^2 \\ 53 & 4 \times 13 + 1 & 7^2 + 2^2 \\ \end{array}$

Also see

• Fermat's Two Squares Theorem: such a prime number is uniquely the sum of two squares

• Definition:Non-Pythagorean Prime