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
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)[2]: n in [2..60] ] ; // Vincenzo Librandi, Mar 16 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jan 19 2014
EXTENSIONS
Terms to a(132) in b-file from Vincenzo Librandi, Mar 16 2019
a(133)-a(658) in b-file from Amiram Eldar, Feb 05 2020
a(659)-a(768) in b-file from Max Alekseyev, Apr 27 2022
STATUS
approved