This site is supported by donations to The OEIS Foundation.



Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 4500 articles have referenced us, often saying "we would not have discovered this result without the OEIS".

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 5


%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 25 18:43 EST 2015. Contains 264418 sequences.