%I A023394
%S A023394 3,5,17,257,641,65537,114689,274177,319489,974849,2424833,6700417,
%T A023394 13631489,26017793,45592577,63766529,167772161,825753601,1214251009,
%U A023394 6487031809,70525124609,190274191361,646730219521,2710954639361
%N A023394 Prime factors of Fermat numbers.
%C A023394 Is it true that this sequence consists of the odd primes p with 2^2^p 
               == 1 (mod p)? (David Wilson, Jul 31 2008). From Max Alekseyev: Yes! 
               If prime p divides Fm = 2^(2^m)+1 then 2^(2^(m+1)) == 1 (mod p) and 
               p is of the form p = k*2^(m+2)+1 > 2^(m+1). Taking squares p - 2^(m+1) 
               times, we have 2^(2^p) == 1 (mod p). On the other hand, if 2^(2^p) 
               == 1 (mod p) for prime p, consider a sequence 2^(2^0), 2^(2^1), 2^(2^2), 
               ..., 2^(2^p). Modulo p this sequence ends with a bunch of 1's but 
               just before the first 1 we must see -1 (as the only other square 
               root of 1 modulo prime p), i.e. for some m, 2^(2^m) == -1 (mod p), 
               implying that p divides Fermat number 2^(2^m) + 1.
%C A023394 Also primes p such that the multiplicative order of 2 (mod p) is a power 
               of 2. A theorem of Lucas states that if m>1 and prime p divides 1+2^2^m 
               (the m-th Fermat number), then p = 1+k*2^(m+2) for some integer k. 
               [From T. D. Noe (noe(AT)sspectra.com), Jan 29 2009]
%C A023394 Wilfrid Keller analyzed the current status of the search for prime factors 
               of Fermat number and stated that all prime factors less than 2^58 
               are now known. He sent me terms a(25) to a(44). [From T. D. Noe (noe(AT)sspectra.com), 
               Feb 01 2009, Feb 03 2009]
%D A023394 M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag 
               NY 2001. [From T. D. Noe (noe(AT)sspectra.com), Jan 29 2009]
%H A023394 T. D. Noe, <a href="b023394.txt">Table of n, a(n) for n=1..44</a> (from 
               Wilfrid Keller)
%H A023394 Wilfrid Keller, <a href="http://www.prothsearch.net/fermat.html">Prime 
               factors k.2^n + 1 of Fermat numbers F_m</a>
%H A023394 R. Munafo, <a href="http://www.mrob.com/pub/math/seq-a023394.html">Prime 
               Factors of Fermat Numbers</a>
%F A023394 a(n) is a prime factor of the Fermat number 1+2^2^A023395(n). [From T. 
               D. Noe (noe(AT)sspectra.com), Feb 01 2009]
%t A023394 Select[Prime[Range[100000]],IntegerQ[Log[2,MultiplicativeOrder[2,# ]]]&] 
               [From T. D. Noe (noe(AT)sspectra.com), Jan 29 2009]
%Y A023394 Cf. A000215.
%Y A023394 Sequence in context: A083213 A056826 A058910 this_sequence A056130 A078726 
               A019434
%Y A023394 Adjacent sequences: A023391 A023392 A023393 this_sequence A023395 A023396 
               A023397
%K A023394 nonn
%O A023394 1,1
%A A023394 David W. Wilson (davidwwilson(AT)comcast.net)