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A023394
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Prime factors of Fermat numbers.
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6
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3, 5, 17, 257, 641, 65537, 114689, 274177, 319489, 974849, 2424833, 6700417, 13631489, 26017793, 45592577, 63766529, 167772161, 825753601, 1214251009, 6487031809, 70525124609, 190274191361, 646730219521, 2710954639361, 2748779069441, 4485296422913, 6597069766657
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OFFSET
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1,1
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COMMENTS
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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.
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. [T. D. Noe, Jan 29 2009]
Wilfrid Keller analyzed the current status of the search for prime factors of Fermat number and stated that all prime factors less than 10^19 are now known. He sent me terms a(25) to a(50). [T. D. Noe, Feb 01 2009, Feb 03 2009, Jan 14 2013]
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REFERENCES
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M. Krizek, F. Luca & L. Somer, 17 Lectures on Fermat Numbers, Springer-Verlag NY 2001.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, Arxiv preprint arXiv:1202.3670, 2012
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..50 (from Wilfrid Keller)
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
R. Munafo, Prime Factors of Fermat Numbers
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FORMULA
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a(n) is a prime factor of the Fermat number 1+2^2^A023395(n). [T. D. Noe, Feb 01 2009]
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MATHEMATICA
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Select[Prime[Range[100000]], IntegerQ[Log[2, MultiplicativeOrder[2, # ]]]&] (* T. D. Noe, Jan 29 2009 *)
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PROG
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(PARI) is(p)=p>2 && Mod(2, p)^lift(Mod(2, znorder(Mod(2, p)))^p)==1 && isprime(p) \\ Charles R Greathouse IV, Feb 04 2013
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CROSSREFS
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Cf. A000215.
Sequence in context: A171271 A056826 A058910 * A176689 A056130 A078726
Adjacent sequences: A023391 A023392 A023393 * A023395 A023396 A023397
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson
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STATUS
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approved
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