On almost universal mixed sums of squares and triangular numbers

Abstract

In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form x 2 + y 2 + 1Oz 2 ; equivalently the form 2x 2 + 5y 2 +4T z represents all integers greater than 1359, where T z denotes the triangular number z(z + 1)/2. Given positive integers a, b, c we employ modular forms and the theory of quadratic forms to determine completely when the general form ax 2 + by 2 + cT z represents sufficiently large integers and to establish similar results for the forms ax 2 + bT y + cT z and aT x + bT y + cT z . Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form 2αx 2 + y 2 + z 2 if and only if all prime divisors of a are congruent to 1 modulo 4. (ii) The form αx 2 + y 2 + T z is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of a is congruent to 1 or 3 modulo 8. (iii) αx 2 +T y + T z is almost universal if and only if all odd prime divisors of a are congruent to 1 modulo 4. (iv) When υ 2 (α) ≠ 3, the form αT x + T y + Tz is almost universal if and only if all odd prime divisors of α are congruent to 1 modulo 4 and υ 2 (α) ≠ 5, 7, ..., where υ 2 (a) is the 2-adic order of α.