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A235366
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Smallest odd prime factor of 3^n - 1.
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7
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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
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3,1
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COMMENTS
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Levi Ben Gerson (1288-1344) 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
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.
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LINKS
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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), 231-234.
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
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FORMULA
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a(4n) = 5 as 3^(4n)-1 = (3^4)^n-1 = 81^n-1 = (80+1)^n-1 == 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 = (730-1)^(1+2n)-1 == (-1)^(1+2n)-1 == -2 (mod 5).
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EXAMPLE
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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.
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MATHEMATICA
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Table[FactorInteger[3^n - 1][[2, 1]], {n, 3, 50}]
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PROG
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(PARI) a(n)=factor(3^n>>valuation(3^n-1, 2))[1, 1] \\ Charles R Greathouse IV, Jan 20 2014
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Jonathan Sondow, Jan 19 2014
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STATUS
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approved
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