Some fascinating results on power sum.

Fixed power sum of positive integers is a well-known problem in number theory. Here we have given a brief history about it by mentioning the contributions of some eminent mathematicians. An internal relation between Σλ and Σλ3 has been traced in an geometric approach and the authors are strongly believing that this geometric approach can give a generalized result for Σλn. Pascal's recurrence relation has been proposed with illustrative examples. Moreover a recurrence relationship is presented by the authors using binomial theorem and they are concluding that any fixed power sum can be easily carried out with this relationship. More importantly by applying binomial theorem we have proved that any fixed power sum is a unique polynomial over rational numbers. Finally an explicit formula has been proposed and this can give any power sum formula. [ABSTRACT FROM AUTHOR]