Proceedings of the Cambridge Philosophical Society, Vol. 41 (June 195), 77-96

[Received 21 November 1944]

Introduction

If f is a real, indefinite, binary quadratic form of discriminant d and if κ(f) is the minimum of |f| taken over all integer values of x,y, not both zero, then it is well known that κ(f) __<__ √(4/5 |d|) and that this is "best possible" result.

In this paper I consider two real binary quadratic forms

f_{1} = a_{1}x^{2}+2b_{1}xy+c_{1}y^{2},
f_{2} =
a_{2}x^{2}+2b_{2}xy+c_{2}y^{2},

which are indefinite and in harmonic relation, i.e. such that

d_{1} = b_{1}^{2}–a_{1}c_{1} > 0,
d_{2} = b_{2}^{2}–a_{2}c_{2} > 0,

and

a_{1}c_{2}–2b_{1}b_{2}+c_{1}c_{2} = 0.

Denote by κ(f_{1}, f_{2}) the minimum of max(| f_{1} |, | f_{2} |) taken over all integer values of x,y, not both zero. I prove that κ(f_{1}, f_{2}) __<__ k(d_{1}, d_{2}), where k is a multiple of either √d_{1} or √d_{2} according to the value of the ratio d_{1}/d_{2}. This is a best possible result. In other words, I find the best possible number k, independent of x,y, such that there exist integers not both zero for which

| f_{1} | __<__ k, | f_{2} | __<__ k.

I find also the pairs of forms for which

max(| f_{1} |, | f_{2} |) = k.