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 A235365 Smallest odd prime factor of 3^n + 1, for n > 1. 8
 5, 7, 41, 61, 5, 547, 17, 7, 5, 67, 41, 398581, 5, 7, 21523361, 103, 5, 2851, 41, 7, 5, 23535794707, 17, 61, 5, 7, 41, 523, 5, 6883, 926510094425921, 7, 5, 61, 41, 18427, 5, 7, 17, 33703, 5, 82064241848634269407, 41, 7, 5, 16921, 76801, 547, 5, 7, 41, 78719947, 5, 61, 17, 7, 5, 3187, 41 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS Levi Ben Gerson (1288-1344) proved that 3^n + 1 = 2^m has no solution in integers if n > 1, by showing that 3^n + l has an odd prime factor. His proof uses remainders after division of powers of 3 by 8 and powers of 2 by 8; see the Lenstra and Peterson links. For an elegant short proof, see the Franklin link. REFERENCES L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731. LINKS Vincenzo Librandi, Table of n, a(n) for n = 2..132 Philip Franklin, Problem 2927, Amer. Math. Monthly, 30 (1923), p. 81. Aaron Herschfeld, The equation 2^x - 3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231-234. Hendrik Lenstra Harmonic Numbers, MSRI, 1998. J. J. O'Connor and E. F. Robertson, Levi ben Gerson, The MacTutor History of Mathematics archive, 2009. Ivars Peterson, Medieval Harmony, Math Trek, MAA, 2012. Wikipedia, Gersonides FORMULA a(2+4n) = 5 as 3^(2+4n) + 1 = (3^2)*(3^4)^n + 1 = 9*81^n + 1 = 9*(80+1)^n + 1 == 9 + 1 == 0 (mod 5). a(3+6n) = 7 as 3^(3+6n) + 1 = (3^3)*(3^6)^n + 1 = 27*729^n + 1 = 27*(728+1)^n + 1 == 27 + 1 == 0 (mod 7), but 27 * 729^n + 1 == 2*(-1)^n + 1 !== 0 (mod 5). EXAMPLE 3^2 + 1 = 10 = 2*5, so a(2) = 5. MATHEMATICA Table[FactorInteger[3^n + 1][[2, 1]], {n, 2, 50}] PROG (MAGMA) [PrimeDivisors(3^n +1): n in [2..60] ] ; // Vincenzo Librandi, Mar 16 2019 CROSSREFS See A235366 for 3^n - 1. Cf. also A003586 (products 2^m * 3^n), A006899, A061987, A108906. Sequence in context: A153376 A235139 A263448 * A189241 A167205 A123781 Adjacent sequences:  A235362 A235363 A235364 * A235366 A235367 A235368 KEYWORD nonn AUTHOR Jonathan Sondow, Jan 19 2014 STATUS approved

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Last modified November 12 19:25 EST 2019. Contains 329078 sequences. (Running on oeis4.)