A080175 Fourth power of primes of the form 4k+1 (A002144).

625, 28561, 83521, 707281, 1874161, 2825761, 7890481, 13845841, 28398241, 62742241, 88529281, 104060401, 141158161, 163047361, 352275361, 492884401, 607573201, 895745041, 1073283121, 1387488001, 1506138481, 2750058481

Offset

1,1

Comments

a(n) is the hypotenuse of four and only four right triangles with integral legs (Fermat). See the Dickson reference, (A) on p. 227.

In 1640 Fermat generalized the 3,4,5 Pythagorean triangle with the theorem: A prime of the form 4k+1 is the hypotenuse of one and only one right triangle with integral legs. The square of a prime of the form 4k+1 is the hypotenuse of two and only two... The cube of three and only three...

References

L. E. Dickson, History of the Theory of Numbers, Volume II, Diophantine Analysis. Carnegie Institution Publication No. 256, Vol II, Washington, DC, 1920, p. 227.

Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A002144(n)^4 = A080109(n)^2, n >= 1.

Product_{n>=1} (1 - 1/a(n)) = A334446. - Amiram Eldar, Dec 02 2022

Example

625 is the hypotenuse of triangles 175, 600, 625; 220, 585, 625; 336, 527, 625; 375, 500, 625.

Maple

seq(p^4, p = select(isprime, [seq(4*k+1, k=1..100)])); # Robert Israel, Jan 14 2015

Mathematica

Select[4 Range[100] + 1, PrimeQ[#] &]^4 (* Vincenzo Librandi, Jun 24 2015 *)

Prog

(PARI) fermat(n) = { for(x=1, n, y=4*x+1; if(isprime(y), print1(y^4, " ")) ) }
(Magma) [a^4: n in [0..40] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Jun 24 2015

Crossrefs

Cf. A002144, A080109, A334446.

Sequence in context: A016972 A156162 A017044 * A017128 A180259 A017224

Adjacent sequences: A080172 A080173 A080174 * A080176 A080177 A080178

Keyword

easy,nonn

Author

Cino Hilliard, Mar 16 2003

Extensions

Edited: name shortened, part of old name as a comment, comment changed, Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 14 2015

Status

approved