

A235366


Smallest odd prime factor of 3^n  1.


7



13, 5, 11, 7, 1093, 5, 13, 11, 23, 5, 797161, 547, 11, 5, 1871, 7, 1597, 5, 13, 23, 47, 5, 11, 398581, 13, 5, 59, 7, 683, 5, 13, 103, 11, 5, 13097927, 1597, 13, 5, 83, 7, 431, 5, 11, 47, 1223, 5, 491, 11, 13, 5, 107, 7, 11, 5, 13, 59, 14425532687, 5, 603901, 683, 13, 5, 11, 7, 221101, 5, 13, 11
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OFFSET

3,1


COMMENTS

Levi Ben Gerson (12881344) proved that 3^n  1 = 2^m has no solution in integers if n > 2, 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.  Sondow
One way to prove it is by the use of congruences. The powers of 3, modulo 80, are 3, 9, 27, 1, 3, 9, 27, 1, 3, 9, 27, 1, ... The powers of 2 are 2, 4, 8, 16, 32, 64, 48, 16, 32, 64, 48, 16, ...  Alonso del Arte, Jan 20 2014


REFERENCES

L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 3..660
P. Franklin, Problem 2927, Amer. Math. Monthly, 30 (1923), p. 81.
A. Herschfeld, The equation 2^x  3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231234.
H. Lenstra Harmonic Numbers, MSRI, 1998.
J. J. O'Connor and E. F. Robertson, Levi ben Gerson, The MacTutor History of Mathematics archive, 2009.
I. Peterson, Medieval Harmony, Math Trek, MAA, 2012.
Wikipedia, Gersonides


FORMULA

a(4n) = 5 as 3^(4n)1 = (3^4)^n1 = 81^n1 = (80+1)^n1 == 0 (mod 5).
a(6+12n) = 7 as 3^(6+12n)1 = (3^6)^(1+2n)1 = 729^(1+2n)1 = (728+1)^(1+2n)1 == 1^(1+2n)1 == 0 (mod 7), but 729^(1+2n)1 = (7301)^(1+2n)1 == (1)^(1+2n)1 == 2 (mod 5).


EXAMPLE

3^3  1 = 26 = 2 * 13, so a(3) = 13.
3^4  1 = 80 = 2^4 * 5, so a(4) = 5.
3^5  1 = 242 = 2 * 11^2, so a(5) = 11.


MATHEMATICA

Table[FactorInteger[3^n  1][[2, 1]], {n, 3, 50}]


PROG

(PARI) a(n)=factor(3^n>>valuation(3^n1, 2))[1, 1] \\ Charles R Greathouse IV, Jan 20 2014


CROSSREFS

See A235365 for 3^n + 1.
Cf. also A003586 (products 2^m * 3^n), A006899, A061987, A108906.
Sequence in context: A098326 A302208 A068662 * A166207 A121230 A299959
Adjacent sequences: A235363 A235364 A235365 * A235367 A235368 A235369


KEYWORD

nonn


AUTHOR

Jonathan Sondow, Jan 19 2014


STATUS

approved



