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

Author

T. D. Noe, Nov 11 2004

Keywords

• nonn

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

Crossrefs

Cf. A099332, A100269, A100270.

Programs

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 *)