A080109 Square of primes of the form 4k+1 (A002144).

25, 169, 289, 841, 1369, 1681, 2809, 3721, 5329, 7921, 9409, 10201, 11881, 12769, 18769, 22201, 24649, 29929, 32761, 37249, 38809, 52441, 54289, 58081, 66049, 72361, 76729, 78961, 85849, 97969, 100489, 113569, 121801, 124609, 139129

Offset: 1

Author

Cino Hilliard, Mar 16 2003

Keywords

• nonn
• easy

Comments

a(n) is the sum of two positive squares in only one way. See the Dickson reference, (B) p. 227.

a(n) is the hypotenuse of two and only two right triangles with integral legs (modulo leg exchange). See the Dickson reference, (A) p. 227.

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...

Examples

			a(7) = 2809 is the hypotenuse of triangles 1241, 2520, 2809 and 1484, 2385, 2809, and only of these.
a(7) = 53^2 = 2809 = 45^2 + (4*7)^2, and this is the only way. - _Wolfdieter Lang_, Jan 13 2015
		

References

• L. E. Dickson, History of the Theory of Numbers, Volume II, Diophantine Analysis. Carnegie Institution Publ. 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
• Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 521.

Crossrefs

Cf. A002144, A080175. - Wolfdieter Lang, Jan 13 2015

Cf. A070079, A070151, A088539, A243380.

Programs

Mathematica

Select[4 Range[96] + 1, PrimeQ]^2 (* Michael De Vlieger, Dec 27 2016 *)

PARI

fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^2" ")) ) }

Formula

a(n) = A002144(n)^2 = A070079(n)^2 + (4*A070151(n))^2, for n >= 1. - Wolfdieter Lang, Jan 13 2015

From Amiram Eldar, Dec 02 2022: (Start)

Product_{n>=1} (1 + 1/a(n)) = A243380

Product_{n>=1} (1 - 1/a(n)) = A088539. (End)

Extensions

Edited: Name changed, part of old name as comment. Comments added and changed. Dickson reference added. - Wolfdieter Lang, Jan 13 2015