# perfect power

The power $m^{n}$ is called a perfect power if $m$ and $n$ are integers both greater than 1. The perfect powers form the ascending order sequence (cf. Sloane’s http://oeis.org/classic/A001597A001597)

 $\displaystyle 4,\,8,\,9,\,16,\,25,\,27,\,32,\,36,\,49,\,64,\,81,\,100,\,121,\,% 125,\,\ldots,$ (1)

i.e.

 $2^{2},\;2^{3},\,3^{2},\;2^{4}=4^{2},\;5^{2},\;3^{3},\;2^{5},\,6^{2},\;7^{2},\;% 2^{6}=4^{3}=8^{2},\;3^{4}=9^{2},\;10^{2},\;11^{2},\;5^{3},\;\ldots$

S. S. Pillai has conjectured in 1945, that if the $i^{\mathrm{th}}$ member of the sequence (1) is denoted by $a_{i}$, then

 $\displaystyle\liminf_{i\to\infty}(a_{i+1}\!-\!a_{i})\;=\;\infty.$ (2)

This does not necessarily that one had  $\lim_{i\to\infty}(a_{i+1}\!-\!a_{i})=\infty$,  since there may always exist little differences $a_{i+1}\!-\!a_{i}$ arbitrarily far from the begin of the sequence (1).

The equation (2) is equivalent (http://planetmath.org/Equivalent3) to the

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

 $x^{m}\!-\!y^{n}\;=\;k$

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:

 $\displaystyle\sum_{m,n=2}^{\infty}\frac{1}{m^{n}}$ $\displaystyle\;=\;\sum_{m=2}^{\infty}\sum_{n=2}^{\infty}\frac{1}{m^{n}}$ $\displaystyle\;=\;\sum_{m=2}^{\infty}\sum_{n=2}^{\infty}\frac{1}{m^{2}}\!\cdot% \!\frac{1}{m^{n-2}}$ $\displaystyle\;=\;\sum_{m=2}^{\infty}\frac{1}{m^{2}}\sum_{n=2}^{\infty}\frac{1% }{m^{n-2}}$ $\displaystyle\;=\;\sum_{m=2}^{\infty}\frac{1}{m^{2}}\sum_{n=0}^{\infty}\left(% \frac{1}{m}\right)^{\!n}$ $\displaystyle\;=\;\sum_{m=2}^{\infty}\frac{1}{m^{2}}\!\cdot\!\frac{1}{1\!-\!% \frac{1}{m}}$ $\displaystyle\;=\;\sum_{m=2}^{\infty}\frac{1}{m(m\!-\!1)}$ $\displaystyle\;=\;\sum_{m=2}^{\infty}\left(\frac{1}{m\!-\!1}-\frac{1}{m}\right)$

The sum of this telescoping series (http://planetmath.org/TelescopingSum) is equal to 1.

## References

• 1 S. S. Pillai:  On $a^{x}\!-\!b^{y}=c$.  – 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