7,076 research outputs found
Almost universal ternary sums of polygonal numbers
For a natural number , generalized -gonal numbers are those numbers of
the form with . In this
paper we establish conditions on for which the ternary sum
is almost universal
A Characterization of almost universal ternary inhomogeneous quadratic polynomials with conductor 2
An integral quadratic polynomial (with positive definite quadratic part) is
called almost universal if it represents all but finitely many positive
integers. In this paper, we provide a characterization of almost universal
ternary quadratic polynomials with conductor 2
On almost universal mixed sums of squares and triangular numbers
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 ,
equivalently the form represents all integers greater than
1359, where denotes the triangular number . Given positive
integers we employ modular forms and the theory of quadratic forms to
determine completely when the general form represents
sufficiently large integers and establish similar results for the forms
and . Here are some consequences of our main
theorems: (i) All sufficiently large odd numbers have the form
if and only if all prime divisors of are congruent to 1 modulo 4. (ii) The
form is almost universal (i.e., it represents sufficiently large
integers) if and only if each odd prime divisor of is congruent to 1 or 3
modulo 8. (iii) is almost universal if and only if all odd prime
divisors of are congruent to 1 modulo 4. (iv) When , the form
is almost universal if and only if all odd prime divisors of
are congruent to 1 modulo 4 and , where is the
2-adic order of .Comment: 35 page
- …