A127317 Numbers n such that (256^n + 1)/257 is prime.

5, 13, 23029

Offset: 1

Author

Alexander Adamchuk, Mar 29 2007

Keywords

• bref
• hard
• more
• nonn

Comments

All terms are primes. Largest currently known prime of the form (2^n + 1)/257 is (256^23029 + 1)/257 found by Donovan Johnson 03/2005. The only currently known prime of the form (2^n + 1)/65537 is (65536^239 + 1)/65537.

From Giuseppe Coppoletta, May 18 2017: (Start)

In general, for any j > 1, if (2^(n*2^j) + 1)/Fj is a prime (where Fj = 2^2^j + 1 is the corresponding Fermat number), then n needs to be prime, as for any odd proper factor q of n, 2^(q*2^j) + 1 is another factor of the numerator. The same for j = 0, apart for the particular value n = 3^2.

For the case j = 4, I checked it again, and (65536^p + 1)/65537 indeed is not a prime at least for 239 < p < 12500, i.e. (2^n + 1)/65537 is not a prime at least up to n = 200000. Any higher upper bound available?

One can also remark that 65536 = 2^16 and 239 = 2^8 - 2^4 - 1. Is there any special reason (see Brennen's link) for that?

I checked also that (2^(p*2^j) + 1)/Fj is never a proper power (in particular it is not a prime power) for j = 0..4 and for any prime p, at least for any exponent p*2^j < 200000.

We can even conjecture that ((Fj-1)^p + 1)/Fj is always squarefree for any odd prime p and for any Fermat number Fj with j >= 0. Note that this is not true if p is not restricted to be a prime, even if p and Fj are coprime, as shown by the following counterexample relative to the case j = 1, f1 = 5: 4^91 + 1 == 0 mod 1093^2. Remark that any such counterexample has to be a Wieferich prime (A001220), but not every Wieferich prime gives a counterexample, as shown by the second known Wieferich prime (3511), which cannot match here because it belongs to A072936.

(End)

Links

• Jack Brennen, Primes of the form (4^p+1)/5^t, Seqfan (Mar 15 2017).
• H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
• H. Lifchitz, Mersenne and Fermat primes field.

Crossrefs

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A057182 = numbers n such that (16^n + 1)/17 is a prime.

Cf. A092559, A073936.

Programs

Mathematica

Do[n=8*Prime[k];f=2^n+1;If[PrimeQ[f/257],Print[{n,n/8}]],{k,1,2570}]