S. S. Pillai has conjectured in 1945, that if the member of the sequence (1) is denoted by , then
This does not necessarily that one had , since there may always exist little differences arbitrarily far from the begin of the sequence (1).
The equation (2) is equivalent (http://planetmath.org/Equivalent3) to the
has only a finite number of solutions where the integers all are greater than 1.
Pillai’s conjecture generalises the Catalan’s conjecture () in which the number of solutions is 1.
The sum of this telescoping series (http://planetmath.org/TelescopingSum) is equal to 1.
- 1 S. S. Pillai: On . – J. Indian math. Soc. 2 (1936).
|Date of creation||2013-03-22 19:15:51|
|Last modified on||2013-03-22 19:15:51|
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