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, 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

Offset: 0

Author

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

Keywords

• nonn
• cons

Comments

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

The first five digits are the same as the first five of A030178 = LambertW(1). - John W. Nicholson, Dec 11 2013

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

Examples

			0.567148130202017714646846875533482564586790249388638206840285221826806766338276...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Octavian Cira, Smarandache's conjecture on consecutive primes, International J. Math. Combin. 4 (2014), pp. 69-91.
• David Lowry-Duda, A short note on gaps between powers of consecutive primes, arXiv:1709.07847 [math.NT], 2017.
• M. L. Perez, Five Smarandache conjectures on primes, Arizona State University, Special Collections.
• F. Smarandache, Conjectures which generalize Andrica's conjecture, arXiv:0707.2584 [math.GM], 2007; Octogon 7:1 (1999), pp. 173-176.
• Eric Weisstein's World of Mathematics, Andrica's Conjecture
• Eric Weisstein's World of Mathematics, Smarandache Constants

Programs

Mathematica

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

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

Extensions

Title improved, incorrect formula deleted, and other edits by M. F. Hasler, Jan 02 2015