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A100268
Primes of the form x^4 + y^4 with x^2 + y^2 and x+y also prime.
3
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
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
KEYWORD
nonn
AUTHOR
T. D. Noe, Nov 11 2004
STATUS
approved