A100268 Primes of the form x^4 + y^4 with x^2 + y^2 and x+y also prime.

2, 17, 97, 257, 641, 1297, 4177, 4721, 12401, 15937, 16561, 38561, 65537, 83537, 89041, 105601, 140321, 160081, 204481, 283937, 284881, 384817, 391921, 411361, 462097, 471617, 531457, 643217, 824641, 838561, 1049201, 1089841, 1342897

Offset

1,1

Comments

The first Mathematica program generates numbers of the form x^4 + y^4 in order of increasing magnitude; it accepts a number when all the x^2^k + y^2^k are prime for k=0,1,2.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Generalized Fermat Number

Mathematica

n=2; pwr=2^n; xmax=2; r=Range[xmax]; num=r^pwr+r^pwr; Table[While[p=Min[num]; x=Position[num, p][[1, 1]]; y=r[[x]]; r[[x]]++; num[[x]]=x^pwr+r[[x]]^pwr; If[x==xmax, xmax++; AppendTo[r, xmax+1]; AppendTo[num, xmax^pwr+(xmax+1)^pwr]]; allPrime=True; k=0; While[k<=n&&allPrime, allPrime=PrimeQ[x^2^k+y^2^k]; k++ ]; !allPrime]; p, {40}]
With[{nn=40}, Select[Union[Transpose[Select[Total/@{#^4, #^2, #}&/@ Tuples[ Range[nn], 2], AllTrue[#, PrimeQ]&]][[1]]], #<=nn^4+1&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 23 2015 *)

Crossrefs

Cf. A099332, A100269, A100270.

Sequence in context: A297727 A358591 A002645 * A163790 A129123 A109724

Adjacent sequences: A100265 A100266 A100267 * A100269 A100270 A100271

Keyword

nonn

Author

T. D. Noe, Nov 11 2004

Status

approved