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June 2020 Waring numbers for diagonal congruences
Ala Alnaser, Todd Cochrane, Misty Ostergaard, Craig Spencer
Rocky Mountain J. Math. 50(3): 825-838 (June 2020). DOI: 10.1216/rmj.2020.50.825

Abstract

For any prime power pn we obtain estimates for the quantity Γ(k,pn), the smallest positive integer s such that for any integers ai with pai, and integer a, the congruence

a 1 x 1 k + + a s x s k a ( mod p n )

is solvable in integers xi, with pxi, 1in. Let k=pek1 with k1|p1, en1 and t1=(p1)k1. For p odd and t1>2 we obtain

Γ ( k , p n ) p p 1 k  and Γ ( k , p n ) ( C log 2 k 1 ) 3 e + 3 k 2 ϕ ( t 1 )

for some constant C. Several other estimates are given for Γ(k,pn) as well as for the related quantities Γ(k,pn), for primitive representations, and Γ0(k,pn), for any representation.

Citation

Download Citation

Ala Alnaser. Todd Cochrane. Misty Ostergaard. Craig Spencer. "Waring numbers for diagonal congruences." Rocky Mountain J. Math. 50 (3) 825 - 838, June 2020. https://doi.org/10.1216/rmj.2020.50.825

Information

Received: 22 August 2018; Revised: 12 November 2019; Accepted: 13 December 2019; Published: June 2020
First available in Project Euclid: 29 July 2020

zbMATH: 07235582
MathSciNet: MR4132612
Digital Object Identifier: 10.1216/rmj.2020.50.825

Subjects:
Primary: 11D79 , 11P05

Keywords: congruences , Waring number

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 3 • June 2020
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