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%I #3 Mar 30 2012 18:40:44
%S 19,179,419,499,643,673,769,883,1153,1409,1459,1889,2003,2083,2131,
%T 2579,2609,2659,2689,2819,3169,3779,3889,3907,4099,4129,4259,4339,
%U 4513,4723,4993,5009,5059,5233,5347,5443,5683,6529,6659,6689,6899,7219,7283,7459
%N Primes which are the sum of four positive 4th powers.
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BiquadraticNumber.html">Biquadratic Number.</a>
%F {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}
%e a(1) = 19 = 2^4 + 1^4 + 1^4 + 1^4 = 16 + 1 + 1 + 1.
%e a(2) = 179 = 3^4 + 3^4 + 2^4 + 1^4 = 81 + 81 + 16 + 1.
%e a(3) = 4^4 + 3^4 + 3^4 + 1^4 = 256 + 81 + 81 + 1.
%t 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]
%Y Cf. A000040, A000583, A003337, A085318.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Dec 31 2007
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