A Pythagorean prime is an odd prime
number of the form
 |
(1)
|
where 
is a positive integer. 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 
is Pythagorean iff
it can be written as
 |
(2)
|
for integers 
. This representation gives the primitive
Pythagorean triple
 |
(3)
|
so every Pythagorean prime occurs as the hypotenuse
of a primitive Pythagorean triangle.
See also
Fermat's 4n+1 Theorem,
Prime Number,
Primitive
Pythagorean Triple,
Pythagorean Triple
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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.Numberphile. "5, 13 and 137 Are
Pythagorean Primes." Jun. 7, 2012. https://www.youtube.com/watch?v=yu_aqA7mw7E.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
