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A100268 Primes of the form x^4 + y^4 with x^2 + y^2 and x+y also prime. 3


%S 2,17,97,257,641,1297,4177,4721,12401,15937,16561,38561,65537,83537,

%T 89041,105601,140321,160081,204481,283937,284881,384817,391921,411361,

%U 462097,471617,531457,643217,824641,838561,1049201,1089841,1342897

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

%C 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.

%H Vincenzo Librandi, <a href="/A100268/b100268.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number </a>

%t 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}]

%t 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 *)

%Y Cf. A099332, A100269, A100270.

%K nonn

%O 1,1

%A _T. D. Noe_, Nov 11 2004

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Last modified July 24 03:01 EDT 2019. Contains 325290 sequences. (Running on oeis4.)