2014-01-01

# On the number of representations of a positive integer as a sum of two binary quadratic forms

## Publication

### Publication

*International Journal of Number Theory , Volume 10 - Issue 6 p. 1395- 1420*

Let N denote the set of positive integers and Z the set of all integers. Let N0 = N ∪ {0}. Let a1x2 + b 1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ N0 as a sum of these two binary quadratic forms is N(a1, b1, c1, a2, b2, c2; n) := card{(x, y, z, t) ∈ Z4 | n = a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2}. When (b1, b2) ≠= (0, 0) we prove under certain conditions on a1, b1, c1, a2, b 2 and c2 that N(a1, b1, c 1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c 1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c 1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ N that N(3, 2, 3, 3, 2, 3; n) = { 0 if n ?1 (mod4), 2σ(N) ifn ?3 (mod4), 0 if n ?2 (mod8), 4σ(N) ifn ? 4,6 (mod8), 8σ(N) ifn ? 8 (mod 16), 24σ(N) ifn ? 0 (mod 16), where N is the largest odd integer dividing n and σ(N) = Σd∈N d|N.

Additional Metadata | |
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Keywords | number of representations, Sum of two binary quadratic forms |

Persistent URL | dx.doi.org/10.1142/S1793042114500353 |

Journal | International Journal of Number Theory |

Citation |
Alaca, S, Pehlivan, L. (Lerna), & Williams, K.S. (2014). On the number of representations of a positive integer as a sum of two binary quadratic forms.
International Journal of Number Theory, 10(6), 1395–1420. doi:10.1142/S1793042114500353 |