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Pillai's Theorem

Pillai's theorem states that consecutive terms in the increasing sequence of nonnegative integer powers of 2 and 3, with 1 included only once, have gaps that tend to infinity nearly as rapidly as the terms themselves (Pillai 1931). Let u_n be the nth term of this sequence, which begins 1, 2, 3, 4, 8, 9, 16, 27, 32, ... (OEIS A006899). More precisely, for every epsilon>0,

u_(n+1)-u_n>u_n^(1-epsilon)

for all sufficiently large n (Hardy 1999, pp. 78-81; Ellison 1970).

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References

Ellison, W. J. "On a Theorem of S. Sivasankaranarayana Pillai." Sém. Théor. Nombres Bordeaux, 1-10, 1970-1971. https://www.numdam.org/item/STNB_1970-1971____A10_0/.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Pillai, S. S. "On the Inequality 0<a^x-b^y<=n." J. Indian Math. Soc. 19, 1-11, 1931.Sloane, N. J. A. Sequence A006899/M0588 in "The On-Line Encyclopedia of Integer Sequences."

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Pillai's Theorem

Cite this as:

Weisstein, Eric W. "Pillai's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PillaisTheorem.html

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