A080169 Numbers that are cubes of primes of the form 4k+1 (A002144).

125, 2197, 4913, 24389, 50653, 68921, 148877, 226981, 389017, 704969, 912673, 1030301, 1295029, 1442897, 2571353, 3307949, 3869893, 5177717, 5929741, 7189057, 7645373, 12008989, 12649337, 13997521, 16974593, 19465109, 21253933

Offset: 1

Author

Cino Hilliard, Mar 16 2003

Keywords

• easy
• nonn

Comments

a(n) is the sum of two squares in exactly two ways (Fermat). See the Dickson reference, (B) on p. 277. - Wolfdieter Lang, Jan 15 2015

a(n) is the hypotenuse of three and only three right triangles with integral arms.

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

See the Dickson reference, (A) on p. 227.

Examples

			a(2) = 2197 is the hypotenuse of the three triangles 825, 2035, 2197; 845, 2028, 2197; 1547, 1560, 2197.
a(2) = 9^2 + 46^2  = 39^2 + 26^2, and these are the only decompositions. - _Wolfdieter Lang_, Jan 15 2015
		

References

• L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 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, A080175, A334425.

Programs

Mathematica

Select[Prime[Range[60]], Mod[#, 4] == 1 &]^3 (* Amiram Eldar, Dec 02 2022 *)

PARI

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^3" ")) ) }

Formula

a(n) = A002144(n)^3, n >= 1.

Product_{n>=1} (1 - 1/a(n)) = A334425. - Amiram Eldar, Dec 02 2022

Extensions

Edited: New name, part of old one now as a comment. Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 15 2015