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

Abstract

Let N denote the set of positive integers and Z the set of all integers. Let N 0 = N ∪ {0}. Let a 1 x 2 + b 1 xy + c 1 y 2 and a 2 z 2 + b 2 zt + c 2 t 2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ N 0 as a sum of these two binary quadratic forms isWhen (b 1 , b 2 ) = (0, 0) we prove under certain conditions on a 1 , b 1 , c 1 , a 2 , b 2 and c 2 that N (a 1 , b 1 , c 1 , a 2 , b 2 , c 2 ; 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 (a 1 , b 1 , c 1 , a 2 , b 2 , c 2 ; n). This determination is carried out explicitly for a number of quaternary quadratic forms a 1 x 2 + b 1 xy + c 1 y 2 + a 2 z 2 + b 2 zt + c 2 t 2 . For example, in Theorem 1.2 we show 1395 Int.

Representation numbers of seven quaternary quadratic forms each in a genus consisting of only two classes

“…The classes of the forms in (3.1) and (3.2) belong to the same genus of discriminant 96 and this genus contains no other classes [17]. (12,12,12,2) showing that the level of Q 1 is 24 . The character associated with Q 1 is…”

“…However, for positive-definite integral nondiagonal quaternary quadratic forms ax 2 + by 2 + cz 2 + dt 2 + exy + f xz + gxt + hyz + jyt + kzt very few have been determined explicitly. The reader can find some in [12]. Even rarer are explicit representation numbers for such forms belonging to a genus containing two or more form classes.…”

Certain quaternary quadratic forms of level 48 and their representation numbers

“…(1, 3, 1), (1,3,2), (1,3,4), (1,3,8), (1,3,16), (1,12,1), (1,12,2), (1,12,4), (1,12,8), χ 0 (1,12,16), (2, 6, 1), (3,4,1), (3,4,2), (3,4,4), (3,4,8), (3,4,16), (4, 12, 1) (1, 6, 1), (1,6,2), (1,6,4), (1,6,8), (1,6,…”

“…(2, 3, 2), (2,3,4), (2,3,8), (2,3,16), (2, 12, 1), (4, 6, 1) (1, 1, 1), (1, 1, 2), (1,1,4), (1,1,8), (1,1,16), (1,4,1), (1,4,2), (1,4,4), χ 12 (1,4,8), (1,4,16), (2, 2, 1), (3, 3, 1), (3, 3, 2), (3,3,4), (3,3,8), (3,3,16), (3, 12, 1), (3,12,2), (3,12,4), (3,12,8), (3,…”

“…There are several works in the literature which consider giving formulas for the representation numbers corresponding to quaternary quadratic forms with coefficients. We list some of them here [1,2,3,4,5,6,7,8,9,19]. In [1], A. Alaca et.…”