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A002586 Smallest prime factor of 2^n + 1.
(Formerly M2385 N0947)
10

%I M2385 N0947

%S 3,5,3,17,3,5,3,257,3,5,3,17,3,5,3,65537,3,5,3,17,3,5,3,97,3,5,3,17,3,

%T 5,3,641,3,5,3,17,3,5,3,257,3,5,3,17,3,5,3,193,3,5,3,17,3,5,3,257,3,5,

%U 3,17,3,5,3,274177,3,5,3,17,3,5,3,97,3,5,3,17,3,5,3,65537,3,5,3,17,3,5

%N Smallest prime factor of 2^n + 1.

%C Conjecture: a(8+48*k) = 257 and a(40+48*k) = 257, where k is a nonnegative integer. - _Thomas König_, Feb 15 2017

%C Conjecture is true: 257 divides 2^(8+48*k)+1 and 2^(40+48*k)+1 but no prime < 257 ever does. Similarly, a(24+48*k) = 97. - _Robert Israel_, Feb 17 2017

%C From _Robert Israel_, Feb 17 2017: (Start)

%C If a(n) = p, there is some m such that a(n+m*j*n) = p for all j.

%C In particular, every member of the sequence occurs infinitely often.

%C a(k*n) <= a(n) for any odd k. (End)

%D J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

%D M. Kraitchik, Recherches sur la Théorie des Nombres, Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 85.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002586/b002586.txt">Table of n, a(n) for n=1..500</a> (using data from A001269)

%H J. Brillhart et al., <a href="http://dx.doi.org/10.1090/conm/022">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

%H Thomas König, <a href="/A002586/a002586.c.txt">C program using gmp for testing the conjectures that a(8+k*48) = 257 and a(40+k*48) = 257</a>

%H E. Lucas, <a href="http://edouardlucas.free.fr/oeuvres/Theorie_des_fonctions_simplement_periodiques.pdf">Théorie des Fonctions Numériques Simplement Périodiques</a>, I", Amer. J. Math., 1 (1878), 184-240, 289-321. See pages 239 and 240.

%H Edouard Lucas, <a href="http://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf">The Theory of Simply Periodic Numerical Functions</a>, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.

%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>

%F a(n) = 3, 5, 3, 17, 3, 5, 3 for n == 1, 2, 3, 4, 5, 6, 7 (mod 8). (Proof. Let n = k*odd with k = 1, 2, or 4. As 2^k = 2, 4, 16 == -1 (mod 3, 5, 17), we get 2^n + 1 = 2^(k*odd) + 1 = (2^k)^odd + 1 == (-1)^odd + 1 == 0 (mod 3, 5, 17). Finally, 2^n + 1 !== 0 (mod p) for prime p < 3, 5, 17, respectively.) - _Jonathan Sondow_, Nov 28 2012

%e a(2^k) = 3, 5, 17, 257, 65537 is the k-th Fermat prime 2^(2^k) + 1 = A019434(k) for k = 0, 1, 2, 3, 4. - _Jonathan Sondow_, Nov 28 2012

%t f[n_] := FactorInteger[2^n + 1][[1, 1]]; Array[f, 100] (* _Robert G. Wilson v_, Nov 28 2012 *)

%Y Cf. A000215, A001269, A002587, A019434, A050922, A093179.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Jul 06 2000

%E Definition corrected by _Jonathan Sondow_, Nov 27 2012

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Last modified March 30 18:46 EDT 2020. Contains 333127 sequences. (Running on oeis4.)