A107961 Pythagorean semiprimes: products of two Pythagorean primes (A002313).

4, 10, 25, 26, 34, 58, 65, 74, 82, 85, 106, 122, 145, 146, 169, 178, 185, 194, 202, 205, 218, 221, 226, 265, 274, 289, 298, 305, 314, 346, 362, 365, 377, 386, 394, 445, 458, 466, 481, 482, 485, 493, 505, 514, 533, 538, 545, 554, 562, 565, 586, 626, 629, 634

Offset

1,1

Comments

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.

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.

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

Links

Table of n, a(n) for n=1..54.

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

Eric Weisstein's World of Mathematics, Semiprime.

Formula

{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}.

Crossrefs

Cf. A001358, A002313, A002330, A002331.

Sequence in context: A038783 A127070 A365861 * A051864 A111153 A265438

Adjacent sequences: A107958 A107959 A107960 * A107962 A107963 A107964

Keyword

easy,nonn

Author

Jonathan Vos Post, Jun 12 2005

Status

approved