A133750 Primes which are the sum of five positive 4th powers.

5, 659, 709, 739, 929, 1283, 1409, 1493, 1523, 1877, 1907, 2099, 2179, 2339, 2689, 2803, 3109, 3187, 3299, 3539, 3733, 3923, 4339, 4357, 5009, 5059, 5443, 5683, 5939, 5987, 6053, 6133, 6529, 7219, 7349, 7459, 7699, 7829, 8419, 8609, 8819, 8849, 9043, 9539

Offset: 1

Author

Jonathan Vos Post, Dec 31 2007

Keywords

• easy
• nonn

Comments

Every positive integer is expressible as a sum of (at most) g(4) = 19 biquadratic numbers (Waring's problem). Davenport (1939) showed that G(4) = 16, meaning that all sufficiently large integers require only 16 biquadratic numbers.

Examples

			a(1) = 5 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 = 1 + 1 + 1 + 1 + 1.
a(2) = 659 = 5^4 + 2^4 + 2^4 + 1^4 + 1^4 = 625 + 16 + 16 + 1 + 1.
a(3) = 709 = 5^4 + 3^4 + 1^4 + 1^4 + 1^4 = 625 + 81 + 1 + 1 + 1.
		

Links

• Eric Weisstein's World of Mathematics, Biquadratic Number.

Crossrefs

Cf. A000040, A000583, A003337, A085318.

Programs

Mathematica

t = Range[9]^4; Select[Union[Plus @@@ Tuples[t, 5]], # < 10^4 && PrimeQ[#] &] (* Giovanni Resta, Jun 20 2016 *)

Formula

{primes} INTERSECTION {a^4 + b^4 + c^4 + d^4 + e^4} = A000040 INTERSECTION {A000583(a) + A000583(b) + A000583(c) + A000583(d) + A000583(e) for a,b,c,d,e > 0}

Extensions

Data corrected by Giovanni Resta, Jun 20 2016