On Graham partitions twisted by the Legendre symbol

Abstract

Abstract We investigate when there is a partition of a positive integer n n , n = f ( λ 1 ) + f ( λ 2 ) + ⋯ + f ( λ ℓ ) , n=f\left({\lambda }_{1})+f\left({\lambda }_{2})+\cdots +f\left({\lambda }_{\ell }), satisfying that 1 = χ p ( λ 1 ) λ 1 + χ p ( λ 2 ) λ 2 + ⋯ + χ p ( λ ℓ ) λ ℓ , 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 χ p {\chi }_{p} is the Legendre symbol modulo prime p p and f ( k ) = k f\left(k)=k or the k k th m m -gonal number with m = 3 m=3 , 4, or 5.