perfect power

The power mn is called a perfect power if m and n are integers both greater than 1. The perfect powers form the ascending order sequenceMathworldPlanetmath (cf. Sloane’s

4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125,, (1)


22, 23, 32, 24=42, 52, 33, 25, 62, 72, 26=43=82, 34=92, 102, 112, 53,

S. S. Pillai has conjectured in 1945, that if the ith member of the sequence (1) is denoted by ai, then

lim infi(ai+1-ai)=. (2)

This does not necessarily that one had  limi(ai+1-ai)=,  since there may always exist little differencesPlanetmathPlanetmath ai+1-ai arbitrarily far from the begin of the sequence (1).

The equation (2) is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( to the

Pillai’s conjecture.  For any positive integer k, the Diophantine equationMathworldPlanetmath


has only a finite number of solutions  (x,y,m,n)  where the integers x,y,m,n all are greater than 1.

Pillai’s conjecture generalises the Catalan’s conjecture (k=1) in which the number of solutions is 1.

The series formed by the inverse numbers of the perfect powers converges absolutely, and its sum may be calculated easily:

m,n=21mn =m=2n=21mn

The sum of this telescoping series ( is equal to 1.


  • 1 S. S. Pillai:  On ax-by=c.  – J. Indian math. Soc. 2 (1936).
Title perfect power
Canonical name PerfectPower
Date of creation 2013-03-22 19:15:51
Last modified on 2013-03-22 19:15:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Definition
Classification msc 11D61
Classification msc 11D45
Classification msc 11B83
Synonym Pillai’s conjecture
Related topic PerfectSquare
Related topic LimitInferior
Related topic DoubleSeries
Related topic SolutionsOfXyYx
Related topic MarshallHallsConjecture