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

Author

Cino Hilliard, Mar 16 2003

Keywords

• easy
• nonn

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...

Examples

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

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

Crossrefs

Cf. A002144, A080109, A334446.

Programs

Magma

[a^4: n in [0..40] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Jun 24 2015

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 *)

PARI

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^4, " ")) ) }

Formula

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

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

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