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Numbers n such that (256^n + 1)/257 is prime.
4

%I #29 Dec 23 2024 14:53:42

%S 5,13,23029

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

%C 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.

%C From _Giuseppe Coppoletta_, May 18 2017: (Start)

%C 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.

%C 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?

%C 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?

%C 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.

%C 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.

%C (End)

%H Jack Brennen, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html">Primes of the form (4^p+1)/5^t, Seqfan (Mar 15 2017)</a>.

%H H. Dubner and T. Granlund, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.html">Primes of the Form (b^n+1)/(b+1)</a>, J. Integer Sequences, 3 (2000), #P00.2.7.

%H H. Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">Mersenne and Fermat primes field</a>.

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

%Y 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.

%Y Cf. A092559, A073936.

%K bref,hard,more,nonn

%O 1,1

%A _Alexander Adamchuk_, Mar 29 2007