A133740 Primes which are the sum of four positive 4th powers.

19, 179, 419, 499, 643, 673, 769, 883, 1153, 1409, 1459, 1889, 2003, 2083, 2131, 2579, 2609, 2659, 2689, 2819, 3169, 3779, 3889, 3907, 4099, 4129, 4259, 4339, 4513, 4723, 4993, 5009, 5059, 5233, 5347, 5443, 5683, 6529, 6659, 6689, 6899, 7219, 7283, 7459

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) = 19 = 2^4 + 1^4 + 1^4 + 1^4 = 16 + 1 + 1 + 1.
a(2) = 179 = 3^4 + 3^4 + 2^4 + 1^4 = 81 + 81 + 16 + 1.
a(3) = 4^4 + 3^4 + 3^4 + 1^4 = 256 + 81 + 81 + 1.
		

Links

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

Crossrefs

Cf. A000040, A000583, A003337, A085318.

Programs

Mathematica

Select[Union[ Flatten[Table[ a^4 + b^4 + c^4 + d^4, {a, 1, 10}, {b, 1, a}, {c, 1, b}, {d, 1, c}]]], PrimeQ]

Formula

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