The gaps in the sequence of perfect powers tend to infinity
~Equivalently~
Every positive integer occurs only finitely many times as a difference of perfect powers
~Generally~
For fixed positive integers \(A,B,C\), the equation \( Ax^n-By^m=C \) has only finitely many solutions \((x,y,m,n)\) for \((m,n)\neq (2,2)\)
| Author | Date Published | |
| Subbayya Sivasankaranarayana Pillai | 1931 |
Pillai proved [1] that the difference \( \left|Ax^n-bx^m\right|\ll x^{\lambda n} \) for \(\lambda<1\), uniformly in \(n\) and \(m\).
The general form of Pillai's Conjecture would follow from the ABC conjecture. [1] [2]
[1] Narkiewicz, Wladyslaw (2011), Rational Number Theory in the 20th Century: From PNT to FLT, Springer Monographs in Mathematics, Springer-Verlag, pp. 253–254, ISBN 978-0-857-29531-6
[2] Schmidt, Wolfgang M. (1996), Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, 1467 (2nd ed.), Springer-Verlag, p. 207, ISBN 3-540-54058-X, Zbl 0754.11020
difference of perfect powers;