Positive Values of Non-homogeneous Indefinite Quadratic Forms of Type (2, 5)

The minimum Γr, n − r of positive values of non-homogeneous indefinite quadratic forms of type (r, n − r) is defined as the infimum of all constants Γ > 0 such that for any indefinite quadratic form Q of type (r, n − r) and determinant D ≠ 0 and any real numbers c1, ..., cn there exist integers x1, ..., xn such that 0 < Q(x1 + c1, ..., xn + cn) < (Γ|D|)1/n. In this paper it is proved that Γ2, 5 = 32, thereby confirming the conjecture of Bambah, Dumir, and Hans-Gill. Also, all the critical forms for which equality is needed are determined.