# Binary quadratic forms – Equivalence and repressentation of integers

If $f,g$ are two binary quadratic forms, $f$ and $g$ are equivalent, if there is an integer matrix $M$ with determinant $\pm1$ such that $G=M^T F M$ where $F,G$ are the matrices that define $f,g$. It is known, that equivalent forms represent the same integers (in fact with same multiplicities). Is the converse also true, i.e., if $f$ and $g$ represent the same integers, are they equivalent? If not, does equivalence hold if also the multiplicities are equal?

#### Solutions Collecting From Web of "Binary quadratic forms – Equivalence and repressentation of integers"

EDIT: I was forgetting two examples with differing discriminants: it is well known, and easy, to show that $x^2 + xy+y^2$ and $x^2 + 3 y^2$ represent exactly the same numbers. With indefinite forms, $x^2 + xy-y^2$ and $x^2 -5 y^2$ represent exactly the same numbers.

ORIGINAL: You seem to be restricting to positive forms. If two forms have the same discriminant and represent the same prime numbers, they are either equivalent or “opposite.” Here equivalent would mean $\det M = 1,$ opposite would mean $\det M = -1.$

Next, what about differing discriminants? There are a finite number of pairs of forms of different discriminants that represent the same primes, or the same odd primes, or wjhat have you. I and Kaplansky wrote an informal article on this. Hendrik Lenstra’s student John Voight, now at Dartmouth, finished the job. Item number 6 at VOIGHT.

Finishing your question is then a matter of checking pairs of forms from Voight’s tables and seeing how they compare on composite numbers. My guess is that perfect agreement is impossible for differing discriminants.

In comparison, for positive forms in three variables, there are infinitely many pairs of forms, of differing discriminants, that represent the same numbers. However, not with the same multiplicities, this is a difficult result due to Alexander Schiemann. I looked into the matter further a year ago, I think I found all pairs of positive ternaries representing the same numbers. No proof of completeness, though.

Schiemann also found an example in four variables, two distinct forms that represent the same numbers with the same multiplicities. It’s in the book by Nipp. I believe the discriminant was 1729. Yes, that’s correct, 1729.