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 A100270 Smallest odd prime of the form x^2^n + y^2^n such that x^2^k + y^2^k is prime for k=0,1,...,n-1. 6
 3, 5, 17, 257, 65537, 43969786939269621239851427694879659964972193373572605276547046131629468448105886917662485986957414531083768961 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The first five terms are the Fermat primes A019434, which are obtained with x=1 and y=2. Can a solution {x,y} be found for any n? The Mathematica program, for each n, generates numbers of the form x^2^n + y^2^n in order of increasing magnitude; it stops when all the x^2^k + y^2^k are prime for k=0,...,n. LINKS Eric Weisstein's World of Mathematics, Generalized Fermat Number EXAMPLE a(5) = 720^32+2669^32 is prime, as are 720^16+2669^16, 720^8+2669^8, 720^4+2669^4, 720^2+2669^2 and 720+2669. MATHEMATICA Table[pwr=2^n; xmax=2; r=Range[xmax]+1; num=(r-1)^pwr+r^pwr; 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, {n, 0, 5}] CROSSREFS Cf. A099332, A100268, A100269. Sequence in context: A000215 A263539 A123599 * A016045 A128336 A278735 Adjacent sequences:  A100267 A100268 A100269 * A100271 A100272 A100273 KEYWORD hard,nice,nonn AUTHOR T. D. Noe, Nov 11 2004 STATUS approved

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Last modified October 18 18:56 EDT 2019. Contains 328197 sequences. (Running on oeis4.)