A291944
a(n) is the least A for which there exists B with 0 < B < A so that A^(2^n) + B^(2^n) is prime.
Original entry on oeis.org
2, 2, 2, 2, 2, 9, 11, 27, 14, 13, 47, 22, 53, 72, 216, 260, 124, 1196, 200
Offset: 0
a(10)=47 corresponds to the prime number 47^1024 + 26^1024, the smallest prime number of the form A^1024 + B^1024 (or more precisely, it minimizes A).
a(14)=216 corresponds to the prime number 216^16384 + 109^16384, a 38248-decimal digit PRP, the smallest prime number of the form A^16384 + B^16384. - _Serge Batalov_, Mar 16 2018
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f[n_] := Monitor[ Block[{a = 2, b}, While[a < Infinity, b = 1 +Mod[a, 2]; While[b < a, If[ PrimeQ[a^2^n + b^2^n], Goto[fini]]; b+=2]; a++]; Label[fini]; {a, b}], {a, b}]; Array[f, 14, 0] (* Robert G. Wilson v, Mar 10 2018 *)
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for(n=0,30,for(a=2,10^100,forstep(b=(a % 2)+1,a-1,2,if(ispseudoprime(a^(2^n)+b^(2^n)),print1(a,", ");next(3)))))
A207261
Primes of the form x^(2*y) + y^(2*x), for x and y > 1.
Original entry on oeis.org
10657, 274200257, 304606801, 92205451297, 22876799984497, 1853020205629057, 59604706692754849, 523348059906214747850254177, 144226335084562589858781936977, 25053659285408524696023221081716801, 100000000000037589973457545958193355601
Offset: 1
10657 is in the sequence because if (x,y) = (3,4), then 3^(2*4) + 4^(2*3) = 6561 + 4096 = 10657.
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a={}; Do[Do[k=x^(2*y)+y^(2*x); If[PrimeQ[k], AppendTo[a,k]], {x,2,y}], {y,2,200}]; Union[a]
A282352
Smallest value of x + y such that x^(2^k) + y^(2^k) is prime for every k = 0..n, where x > y are nonnegative integers.
Original entry on oeis.org
2, 3, 3, 3, 3, 3389, 63559
Offset: 0
a(0) = 2 because 2^(2^0) + 0^(2^0) = 2.
a(1) = 3 because 2^(2^0) + 1^(2^0) = 3 and 2^(2^1) + 1^(2^1) = 5 are prime.
a(2) = 3 because 2^(2^0) + 1^(2^0) = 3, 2^(2^1) + 1^(2^1) = 5, and 2^(2^2) + 1^(2^2) = 17 are prime.
a(n) x y
----- ----- -----
2 2 0
3 2 1
3 2 1
3 2 1
3 2 1
3389 2669 720
63559 34559 29000
A123646
Smallest generalized Fermat prime of the form a^(2^n) + b^(2^n), where bases a,b>1; or -1 if no such prime exists.
Original entry on oeis.org
5, 13, 97, 2070241, 4338014017, 3512911982806776822251393039617, 4457915690803004131256192897205630962697827851093882159977969339137, 1638935311392320153195136107636665419978585455388636669548298482694235538906271958706896595665141002450684974003603106305516970574177405212679151205373697500164072550932748470956551681
Offset: 0
Cf.
A000215 (Fermat numbers: 2^(2^n) + 1),
A019434 (Fermat primes of the form 2^(2^n) + 1).
Cf.
A111635 (allows one of a,b to be 1).
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Do[f=Min[Select[Take[Union[ Flatten[ Table[ i^(2^n) + j^(2^n), {i, 2, 300}, {j, 2, 300} ] ] ],500],PrimeQ]];Print[{n,f}],{n,0,7}]
A269834
Primes p of the form 2^k + 9*(-1)^k - 8.
Original entry on oeis.org
2, 5, 17, 257, 65537
Offset: 1
A343121
a(n) is the least A for which there exists B with 0 < B < A so that A^(2^k) + B^(2^k) is prime for k = 0, 1, ..., n.
Original entry on oeis.org
2, 2, 2, 2, 2, 2669, 34559, 26507494, 3242781025
Offset: 0
For n=5, the six numbers 2669 + 720, 2669^2 + 720^2, 2669^4 + 720^4, 2669^8 + 720^8, 2669^16 + 720^16, and 2669^32 + 720^32 are all prime, and (A,B) = (2669,720) is the least pair with this property, so a(5)=2669.
For n=6, (A,B) = (34559,29000).
For n=7, (A,B) = (26507494,6329559).
For n=8, (A,B) = (3242781025,1554825312).
- Yves Gallot, xgfp8, software for calculating this sequence.
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a(n)=for(A=1, oo, for(B=1, A-1, for(k=0, n, !ispseudoprime(A^(2^k)+B^(2^k)) && next(2)); return(A)))
Showing 1-6 of 6 results.
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