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A038458 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. 5


%S 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,

%T 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,

%U 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

%N 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.

%C Generalizes Andrica's conjecture p(n+1)^(1/2) - p(n)^(1/2) < 1 to p(n+1)^c - p(n)^c < 1 if c is less than this number.

%C Is this constant rational or irrational? I conjecture it is irrational. - Sukanto Bhattacharya (susant5au(AT)yahoo.com.au), Apr 28 2008

%C The first five digits are the same as the first five of A030178 = LambertW(1). - _John W. Nicholson_, Dec 11 2013

%H Harry J. Smith, <a href="/A038458/b038458.txt">Table of n, a(n) for n = 0..20000</a>

%H Octavian Cira, <a href="http://mathcombin.com/upload/file/20150127/1422320940239094100.pdf#page=74">Smarandache's conjecture on consecutive primes</a>, International J. Math. Combin. 4 (2014), pp. 69-91.

%H David Lowry-Duda, <a href="https://arxiv.org/abs/1709.07847">A short note on gaps between powers of consecutive primes</a>, arXiv:1709.07847 [math.NT], 2017.

%H M. L. Perez, <a href="http://www.gallup.unm.edu/~smarandache/conjprim.txt">Five Smarandache conjectures on primes</a>, Arizona State University, Special Collections.

%H F. Smarandache, <a href="http://arxiv.org/abs/0707.2584">Conjectures which generalize Andrica's conjecture</a>, arXiv:0707.2584 [math.GM], 2007; Octogon 7:1 (1999), pp. 173-176.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AndricasConjecture.html">Andrica's Conjecture</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheConstants.html">Smarandache Constants</a>

%e 0.567148130202017714646846875533482564586790249388638206840285221826806766338276...

%t RealDigits[x/.FindRoot[127^x-113^x==1,{x,0.5},WorkingPrecision->150]][[1]] (* _Harvey P. Dale_, Oct 24 2017 *)

%o (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

%K nonn,cons

%O 0,1

%A M. I. Petrescu (mipetrescu(AT)yahoo.com)

%E Title improved, incorrect formula deleted, and other edits by _M. F. Hasler_, Jan 02 2015

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Last modified September 20 08:13 EDT 2019. Contains 327214 sequences. (Running on oeis4.)