On Graham partitions twisted by the Legendre symbol

Abstract

We investigate when there is a partition of a positive integer nn, n=f(λ1)+f(λ2)+⋯+f(λℓ),n=f\left({\lambda }_{1})+f\left({\lambda }_{2})+\cdots +f\left({\lambda }_{\ell }), satisfying that 1=χp(λ1)λ1+χp(λ2)λ2+⋯+χp(λℓ)λℓ,1=\frac{{\chi }_{p}\left({\lambda }_{1})}{{\lambda }_{1}}+\frac{{\chi }_{p}\left({\lambda }_{2})}{{\lambda }_{2}}+\cdots +\frac{{\chi }_{p}\left({\lambda }_{\ell })}{{\lambda }_{\ell }}, where χp{\chi }_{p} is the Legendre symbol modulo prime pp and f(k)=kf\left(k)=k or the kkth mm-gonal number with m=3m=3, 4, or 5.

Keywords

• graham partition
• sum of reciprocals
• quadratic twist
• legendre symbol
• polygonal number
• 11p81
• 05a17