A100270 - OEIS

%I #9 Feb 21 2023 06:17:52

%S 3,5,17,257,65537,

%T 43969786939269621239851427694879659964972193373572605276547046131629468448105886917662485986957414531083768961

%N 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.

%C 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.

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

%e 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.

%t 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}]

%Y Cf. A099332, A100268, A100269.

%K hard,nice,nonn

%O 0,1

%A _T. D. Noe_, Nov 11 2004