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%I #251 Apr 03 2023 10:36:09
%S 3,5,17,257,65537
%N Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0.
%C It is conjectured that there are only 5 terms. Currently it has been shown that 2^(2^k) + 1 is composite for 5 <= k <= 32 (see Eric Weisstein's Fermat Primes link). - _Dmitry Kamenetsky_, Sep 28 2008
%C No Fermat prime is a Brazilian number. So Fermat primes belong to A220627. For proof see Proposition 3 page 36 in "Les nombres brésiliens" in Links. - _Bernard Schott_, Dec 29 2012
%C This sequence and A001220 are disjoint (see "Other theorems about Fermat numbers" in Wikipedia link). - _Felix Fröhlich_, Sep 07 2014
%C Numbers n > 1 such that n * 2^(n-2) divides (n-1)! + 2^(n-1). - _Thomas Ordowski_, Jan 15 2015
%C From _Jaroslav Krizek_, Mar 17 2016: (Start)
%C Primes p such that phi(p) = 2*phi(p-1); primes from A171271.
%C Primes p such that sigma(p-1) = 2p - 3.
%C Primes p such that sigma(p-1) = 2*sigma(p) - 5.
%C For n > 1, a(n) = primes p such that p = 4 * phi((p-1) / 2) + 1.
%C Subsequence of A256444 and A256439.
%C Conjectures:
%C 1) primes p such that phi(p) = 2*phi(p-2).
%C 2) primes p such that phi(p) = 2*phi(p-1) = 2*phi(p-2).
%C 3) primes p such that p = sigma(phi(p-2)) + 2.
%C 4) primes p such that phi(p-1) + 1 divides p + 1.
%C 5) numbers n such that sigma(n-1) = 2*sigma(n) - 5. (End)
%C Odd primes p such that ratio of the form (the number of nonnegative m < p such that m^q == m (mod p))/(the number of nonnegative m < p such that -m^q == m (mod p)) is a divisor of p for all nonnegative q. - _Juri-Stepan Gerasimov_, Oct 13 2020
%C Numbers n such that tau(n)*(number of distinct ratio (the number of nonnegative m < n such that m^q == m (mod n))/(the number of nonnegative m < n such that -m^q == m (mod n))) for nonnegative q is equal to 4. - _Juri-Stepan Gerasimov_, Oct 22 2020
%C The numbers of primitive roots for the five known terms are 1, 2, 8, 128, 32768. - _Gary W. Adamson_, Jan 13 2022
%C Prime numbers such that every residue is either a primitive root or a quadratic residue. - _Keith Backman_, Jul 11 2022
%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see Table 1, p. 458.
%D R. K. Guy, Unsolved Problems in Number Theory, A3.
%D Hardy and Wright, An Introduction to the Theory of Numbers, bottom of page 18 in the sixth edition, gives an heuristic argument that this sequence is finite.
%H Anas AbuDaqa, Amjad Abu-Hassan, and Muhammad Imam, <a href="https://arxiv.org/abs/2006.08444">Taxonomy and Practical Evaluation of Primality Testing Algorithms</a>, arXiv:2006.08444 [cs.CR], 2020.
%H Cyril Banderier, <a href="https://web.archive.org/web/20060217222523/http://algo.inria.fr/banderier/Recipro/node35.html">Pepin's Criterion For Fermat Numbers</a> (in French)
%H Kent D. Boklan and John H. Conway, <a href="http://arxiv.org/abs/1605.01371">Expect at most one billionth of a new Fermat Prime!</a>, arXiv:1605.01371 [math.NT], 2016.
%H P. Bruillard, S.-H. Ng, E. Rowell, and Z. Wang, <a href="http://arxiv.org/abs/1310.7050">On modular categories</a>, arXiv:1310.7050 [math.QA], 2013-2015.
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/FermatNumber.html">Fermat number</a>
%H R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H Wilfrid Keller, <a href="http://www.prothsearch.com/fermat.html">Prime factors k.2^n + 1 of Fermat numbers F_m</a>
%H P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, <a href="https://arxiv.org/abs/2201.06636">On digital sequences associated with Pascal's triangle</a>, arXiv:2201.06636 [math.NT], 2022.
%H Romeo Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv:1202.3670 [math.HO], 2012. - _N. J. A. Sloane_, Jun 13 2012
%H Romeo Meštrović, <a href="https://www.researchgate.net/publication/329844912_GOLDBACH-TYPE_CONJECTURES_ARISING_FROM_SOME_ARITHMETIC_PROGRESSIONS">Goldbach-type conjectures arising from some arithmetic progressions</a>, University of Montenegro, 2018.
%H Romeo Meštrović, <a href="https://arxiv.org/abs/1901.07882">Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes</a>, arXiv:1901.07882 [math.NT], 2019.
%H Salah Eddine Rihane, Chèfiath Awero Adegbindin, and Alain Togbé, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Togbe/togbe16.html">Fermat Padovan And Perrin Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.6.2.
%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/1701.04741">New Congruences and Finite Difference Equations for Generalized Factorial Functions</a>, arXiv:1701.04741 [math.CO], 2017.
%H Bernard Schott, <a href="http://quadrature.info/">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38. <a href="/A125134/a125134.pdf">Local copy</a>, included here with permission from the editors of Quadrature.
%H Yaryna Stelmakh, <a href="https://arxiv.org/abs/2101.04676">Homeomorphisms of the space of non-zero integers with the Kirch topology</a>, arXiv:2101.04676 [math.GN], 2020.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatPrime.html">Fermat Prime</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PepinsTest.html">Pepin's Test</a>
%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Fermat_prime">Fermat number</a>
%H Haifeng Xu, <a href="https://arxiv.org/abs/1601.06509">The largest cycles consist by the quadratic residues and Fermat primes</a>, arXiv:1601.06509 [math.NT], 2016.
%F a(n+1) = A180024(A049084(a(n))). - _Reinhard Zumkeller_, Aug 08 2010
%F a(n) = 1 + A001146(n-1), if 1 <= n <= 5. - _Omar E. Pol_, Jun 08 2018
%t Select[Table[2^(2^n) + 1, {n, 0, 4}], PrimeQ] (* _Vladimir Joseph Stephan Orlovsky_, Apr 29 2008 *)
%o (Magma) [2^(2^n)+1 : n in [0..4] | IsPrime(2^(2^n)+1)]; // _Arkadiusz Wesolowski_, Jun 09 2011
%o (PARI) for(i=0,10, isprime(2^2^i+1) && print1(2^2^i+1,", ")) \\ _M. F. Hasler_, Nov 21 2009
%o (Sage) [2^(2^n)+1 for n in (0..4) if is_prime(2^(2^n)+1)] # _G. C. Greubel_, Mar 07 2019
%Y Cf. A000215, A001146, A159611, A220627, A220570.
%Y Subsequence of A147545 and of A334101. Cf. also A333788, A334092.
%Y Cf. A045544.
%K nonn,hard,nice
%O 1,1
%A _N. J. A. Sloane_, _David W. Wilson_
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