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Shanks' Conjecture

Let p(n) be the first prime which follows a prime gap of n between consecutive primes. Shanks' conjecture holds that

p(n)∼exp(sqrt(n)).

Wolf conjectures a slightly different form

p(n)∼sqrt(n)exp(sqrt(n)),

which agrees better with numerical evidence.

See also

Prime Difference Function, Prime Gaps

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References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 21, 1994.Rivera, C. "Problems & Puzzles: Conjecture 009.-Shanks' Conjecture." http://www.primepuzzles.net/conjectures/conj_009.htm.Shanks, D. "On Maximal Gaps Between Successive Primes." Math. Comput. 18, 646-651, 1964.Wolf, M. "First Occurrence of a Given Gap Between Consecutive Primes." http://www.ift.uni.wroc.pl/~mwolf/.

Referenced on Wolfram|Alpha

Shanks' Conjecture

Cite this as:

Weisstein, Eric W. "Shanks' Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ShanksConjecture.html

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