On almost universal mixed sums of squares and triangular numbers

TL;DR

This paper characterizes when mixed sums of squares and triangular numbers are almost universal, using modular forms and quadratic form theory, extending previous results and providing explicit prime divisor conditions.

Contribution

It completely determines conditions for various mixed quadratic-triangular forms to represent all sufficiently large integers, generalizing earlier work under the Riemann hypothesis.

Findings

Form $2ax^2 + y^2 + z^2$ represents all large odd numbers if prime divisors of $a$ are ≡ 1 mod 4.

Form $ax^2 + y^2 + T_z$ is almost universal if odd prime divisors of $a$ are ≡ 1 or 3 mod 8.

Form $ax^2 + T_y + T_z$ is almost universal if all odd prime divisors of $a$ are ≡ 1 mod 4.

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 $x^2+y^2+10z^2$, 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 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 $2ax^2+y^2+z^2$ if and only if all prime divisors of $a$ are congruent to 1 modulo 4. (ii) The form $ax^2+y^2+T_z$ 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) $ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$ are congruent to 1 modulo 4. (iv) When $v_2(a)\not=3$, the form $aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$ are congruent to 1 modulo 4 and $v_2(a)\not=5,7,...$, where $v_2(a)$ is the 2-adic order of $a$.