Indefinite binary forms representing the same numbers

Abstract

It is an old problem whether two binary quadratic forms representing the same numbers are equivalent. Legendre asserted without proof that two positive definite forms representing the same numbers are equivalent. In 1895, Bauer(1) proved Legendre's assertion for primitive forms with the same discriminant. In 1938 Delone (2) proved that two positive definite binary forms with real coefficients are equivalent if they represent the same numbers, with the sole exception of the pair equivalent toand their scalings. Watson (3) rediscovered this result in 1979 and asked the same question for indefinite binary forms.