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Stephens' Constant


Let a and b be nonzero integers such that a^mb^n!=1 (except when m=n=0). Also let T(a,b) be the set of primes p for which p|(a^k-b) for some nonnegative integer k. Then assuming the generalized Riemann hypothesis, Stephens (1976) showed that the density of T(a,b) relative to the primes is a rational multiple of

 C_(Stephens)=product_(j=1)^infty(1-(p_j)/(p_j^3-1))=0.5759599688...

(OEIS A065478), where p_j is the jth prime (Finch 2003).


See also

Artin's Constant

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References

Finch, S. R. "Artin's Constant." §2.4 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 104-110, 2003.Moree, P. "Approximation of Singular Series and Automata." Submitted to Manuscripta Math. 101, 385-399, 2000.Moree, P. and Stevenhagen, P. "A Two-Variable Artin Conjecture." J. Number Th. 85, 291-304, 2000.Niklasch, G. "Some Number-Theoretical Constants." http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml.Sloane, N. J. A. Sequence A065478 in "The On-Line Encyclopedia of Integer Sequences."Stephens, P. J. "Prime Divisor of Second-Order Linear Recurrences, I." J. Number Th. 8, 313-332, 1976.

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Stephens' Constant

Cite this as:

Weisstein, Eric W. "Stephens' Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StephensConstant.html

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