Agnes Scott College

Kathleen Ollerenshaw

The Minima of a Pair of Indefinite, Harmonic, Binary Quadratic Forms
Proceedings of the Cambridge Philosophical Society, Vol. 41 (June 195), 77-96

[Received 21 November 1944]


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

f1 = a1x2+2b1xy+c1y2, f2 = a2x2+2b2xy+c2y2,

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

d1 = b12–a1c1 > 0,    d2 = b22–a2c2 > 0,


a1c2–2b1b2+c1c2 = 0.

Denote by κ(f1, f2) the minimum of max(| f1 |, | f2 |) taken over all integer values of x,y, not both zero. I prove that κ(f1, f2) < k(d1, d2), where k is a multiple of either √d1 or √d2 according to the value of the ratio d1/d2. 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

| f1 | < k,    | f2 | < k.

I find also the pairs of forms for which

max(| f1 |, | f2 |) = k.