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.
Original entry on oeis.org
3, 5, 17, 257, 65537, 43969786939269621239851427694879659964972193373572605276547046131629468448105886917662485986957414531083768961
Offset: 0
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.
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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}]
A100268
Primes of the form x^4 + y^4 with x^2 + y^2 and x+y also prime.
Original entry on oeis.org
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
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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 *)
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