1,1

Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique manner (up to the order of addends) in the form x^2 + y^2 for integer x and y iff p = 1 (mod 4) or p = 2 (which is a degenerate case with x = y = 1). The theorem was stated by Fermat, but the first published proof was by Euler.

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.

Seroul, R. "Prime Number and Sum of Two Squares." Section 2.11 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 18-19, 2000.

Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.

Eric Weisstein's World of Mathematics, Semiprime.

{a(n)} = {p*q: p and q both elements of A002313} = {p*q: p and q both of form m^2 + n^2 for integers m, n}.

Cf. A001358, A002313, A002330, A002331.

Sequence in context: A038783 A127070 A365861 * A051864 A111153 A265438

Adjacent sequences: A107958 A107959 A107960 * A107962 A107963 A107964

easy,nonn

Jonathan Vos Post, Jun 12 2005

approved