Math Graduate Student Seminar: "An investigation of power sums of integers"
Nicholas Newsome, CSU Fresno
Friday, April 7, 2017
1:10–2 p.m.
Location: Surge Building 284
Parking Information
Category: Seminar
Description:
Sums of powers of integers have been studied extensively for many centuries. The Pythagoreans, Archimedes, Fermat, Pascal, Bernoulli, Faulhaber, and other mathematicians have discovered formulas for sums of powers of the first n natural numbers. Among these is Faulhaber's wellknown formula which expresses the power sums as polynomials whose coefficients involve Bernoulli numbers.
In this talk, we sketch an elementary proof that for each natural number p, the sum of pth powers of the first n natural numbers can be expressed as a polynomial in n of degree p 1. We also present a novel identity involving Bernoulli numbers and use it to show symmetry of these polynomials. In addition, we make a few conjectures regarding the roots of these polynomials, and speculate on the asymptotic behavior of their graphs. Finally, we study the remainders of the power sums upon division by integers. In particular, we generalize a wellknown result on congruence of Sp(n) from prime n to any power of prime and state periodicity properties of Sp(n) mod k.
Additional Information: Math Seminars
Open to: General Public
Admission: Free
Sponsor: Mathematics
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