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 $x^2+y^2+10z^2$, 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 $a{x}^2+b{y}^2+c{T}_z$ represents sufficiently large integers and establish similar results for the forms $a{x}^2+b{T}_y+c{T}_z$ and $a{T}_x+b{T}_y+c{T}_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $2a{x}^2+y^2+z^2$ if and only if all prime divisors of $a$ are congruent to 1 modulo 4. (ii) The form $a{x}^2+y^2+T_z$ 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) $a{x}^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$ are congruent to 1 modulo 4. (iv) When $v_2(a)\ne 3$, the form $a{T}_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$ are congruent to 1 modulo 4 and $v_2(a)\ne 5,7,...$, where $v_2(a)$ is the 2-adic order of $a$.

35 pages

References in corpus (2)

• On sums of primes and triangular numbers
• Mixed sums of squares and triangular numbers (III)