Counting the number of solutions to certain infinite Diophantine equations

2019/10/17 by Zhou, Nian Hong, Sun, Yalin
#11P99 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11D45 #Secondary: 11P82

paper · doi:10.48550/arxiv.1910.07884

Abstract

Let r, v, n be positive integers. This paper investigate the number of solutions s_r,v(n) of the following infinite Diophantine equations n=1^r⋅ |k₁|^v+2^r⋅ |k₂|^v+3^r⋅ |k₃|^v+…, for \bf k=(k₁,k₂,k₃,…)∈ℤ^∞. For each (r,v)∈ℕ×\1,2\, a generating function and some asymptotic formulas of s_r,v(n) are established.

Related

• Solving an infinite number of purely exponential Diophantine equations with four terms
• Quantitative indicators of the solutions of Diophantine equations and systems in the domain of the natural numbers
• Determination of the asymptotic behavior of the number of natural solutions for certain types of diagonal Diophantine equations
• Remark on a Paper by Izadi and Baghalaghdam about Cubes and Fifth Powers Sums
• On the Diophantine equation σ₂(X_n)=σ_n(X_n)