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A038458
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Decimal expansion of the solution to 127^x - 113^x = 1. This is the smallest x such that q^x - p^x = 1 for two successive primes p, q.
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6
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5, 6, 7, 1, 4, 8, 1, 3, 0, 2, 0, 2, 0, 1, 7, 7, 1, 4, 6, 4, 6, 8, 4, 6, 8, 7, 5, 5, 3, 3, 4, 8, 2, 5, 6, 4, 5, 8, 6, 7, 9, 0, 2, 4, 9, 3, 8, 8, 6, 3, 8, 2, 0, 6, 8, 4, 0, 2, 8, 5, 2, 2, 1, 8, 2, 6, 8, 0, 6, 7, 6, 6, 3, 3, 8, 2, 7, 6, 9, 2, 1, 5, 0, 8, 8, 6, 9, 7, 3, 8, 5, 3, 6, 4, 2, 6, 4, 4
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OFFSET
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0,1
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COMMENTS
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Generalizes Andrica's conjecture prime(n+1)^(1/2) - prime(n)^(1/2) < 1 to prime(n+1)^c - prime(n)^c < 1 if c is less than this number.
Is this constant rational or irrational? I conjecture it is irrational. - Sukanto Bhattacharya (susant5au(AT)yahoo.com.au), Apr 28 2008
Although the description of the sequence defines it as "the smallest x" with a certain property, this is conjectured, not yet proven. Numerical evidence supports the conjecture. - Hal M. Switkay, Jun 02 2021
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LINKS
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EXAMPLE
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0.567148130202017714646846875533482564586790249388638206840285221826806766338276...
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MATHEMATICA
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RealDigits[x/.FindRoot[127^x-113^x==1, {x, 0.5}, WorkingPrecision->150]][[1]] (* Harvey P. Dale, Oct 24 2017 *)
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PROG
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(PARI) default(realprecision, 20080); x=solve(x=.5, .6, 127^x-113^x-1); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b038458.txt", n, " ", d)); \\ Harry J. Smith, Apr 13 2009
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CROSSREFS
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KEYWORD
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AUTHOR
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M. I. Petrescu (mipetrescu(AT)yahoo.com)
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EXTENSIONS
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Title improved, incorrect formula deleted, and other edits by M. F. Hasler, Jan 02 2015
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STATUS
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approved
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