Representations by certain octonary quadratic forms with coefficients 1, 5, or 25

Barış Kendirli

doi:10.28919/jmcs/3790

Abstract

It is an important objective to determine the number of representations of a positive integer by certain quadratic forms in number theory. Formulae for $% N(1^{2i},2^{2j},3^{2k},6^{2l};n)$ for the nine octonary quadratic forms appear in the literature, whose coefficients are $1,2,3$ and $6$. Moreover, the formulae for $N(1^{i},3^{j},9^{k};n)$ for several octonary quadratic forms have been given by Alaca. Here, we determine formulae, for $N(1^{i},5^{j},25^{k};n)$ for several octonary quadratic forms.

Highlights

It is interesting and important to determine explicit formulas of the representation number of positive definite quadratic forms.The work on representation number card{(x1, x2) ∈ Z2|n = x12 + x22} of quadratic form x2 + y2 has been started by Fermat in 1640
It is an important objective to determine the number of representations of a positive integer by certain quadratic forms in number theory
Formulae for N(12i, 22 j, 32k, 62l; n) for the nine octonary quadratic forms appear in the literature, whose coefficients are 1, 2, 3 and 6

Summary

Introduction

It is interesting and important to determine explicit formulas of the representation number of positive definite quadratic forms. The work on representation number card{(x1, x2) ∈ Z2|n = x12 + x22} of quadratic form x2 + y2 has been started by Fermat in 1640. First systematic treatment of binary quadratic forms is due to Legendre. Afterwards it was advanced by Jacobi, with the proof of. There are many more representation number formulas obtained for quadratic forms. The formulae for N(1i, 3 j, 9k; n) for several octonary quadratic forms have been given by Alaca [5]. We determine formulae, for N(1i, 5 j, 25k; n) for several octonary quadratic forms

Preliminaries

Main results

CONCLUSION