TOPICS
Search

Pythagorean Prime

A Pythagorean prime is an odd prime number of the form

p=4n+1.
(1)

The first few are 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, ... (OEIS A002144).

By Fermat's 4n+1 theorem, a prime p is Pythagorean if and only if it can be written as

p=a^2+b^2,
(2)

for integers a>b>0. This representation gives the primitive Pythagorean triple

(a^2-b^2,2ab,p),
(3)

so every Pythagorean prime occurs as the hypotenuse of a primitive right triangle with integer side lengths.

See also

Fermat's 4n+1 Theorem, Prime Number, Primitive Pythagorean Triple, Pythagorean Triple

Explore with Wolfram|Alpha

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 146-147 and 220-223, 1996.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 13 and 219, 1979.Sloane, N. J. A. Sequence A002144 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Pythagorean Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PythagoreanPrime.html

Subject classifications