A235365 Smallest odd prime factor of 3^n + 1, for n > 1.

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

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

Table of n, a(n) for n = 2..768

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

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

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