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.

3, 5, 17, 257, 65537, 43969786939269621239851427694879659964972193373572605276547046131629468448105886917662485986957414531083768961

Offset: 0

Author

T. D. Noe, Nov 11 2004

Keywords

• hard
• nice
• nonn

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.

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Crossrefs

Cf. A099332, A100268, A100269.

Programs

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}]