Intereting Posts

Palindromic Numbers – Pattern “inside” Prime Numbers?
$\ln|x|$ vs.$\ln(x)$? When is the $\ln$ antiderivative marked as an absolute value?
expected number of edges in a random graph
Definite Integral of square root of polynomial
Determination of the last three digits of $2014^{2014}$
How can I show that $\left|\sum_{n=1}^\infty\frac{x}{n^2+x^2}\right|\leq\frac{\pi}{2}$ for any $x\in{\bf R}$?
subgroups of finitely generated groups with a finite index
A question regarding normal field extensions and Galois groups
Give an example of a bounded, non-convergent real sequence $(a_n)$ s.t. $a_n-a_{n-1}\rightarrow 0$
Binomial Congruence
Prove $( \lnot C \implies \lnot B) \implies (B \implies C)$ without the Deduction Theorem
Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?
Motivating implications of the axiom of choice?
Havel & Hakimi degree sequence theory
Intuitive/heuristic explanation of Polya's urn

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?

- Integer solutions to $\prod\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}x_i^2$
- Given a, b How many solutions exists for x, such that: $a \bmod{x}=b $
- Problem Heron of Alexandria.
- how to prove this extended prime number theorem?
- How do you proof that the simply periodic continuous fraction is palindromic for the square root of positive primes?
- The product of two numbers that can be written as the sum of two squares
- Deciding whether $2^{\sqrt2}$ is irrational/transcendental
- writing $pq$ as a sum of squares for primes $p,q$
- Discriminant of splitting field
- A low-degree polynomial $g_{a,b}(x)$ which has a zero $x\in\mathbb N$ for any square numbers $a,b$?

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.

- If $F$ be a field, then $F$ is a principal ideal domain. Does $F$ have to be necessarily a field?
- Sum of Cosine/Exponential
- $f$ is continuous at $a$ iff for each subset $A$ of $X$ with $a\in \bar A$, $f(a)\in \overline{ f(A)}$.
- Why do we say the harmonic series is divergent?
- Treating shocks with conservation laws
- A net version of dominated convergence?
- Existence of the absolute value of the limit implies that either $f \ $ or $\bar{f} \ $ is complex-differentiable
- Simple Nonlinear Differential Equation
- A function for which one-sided limits are zero and infinity
- Why convolution regularize functions?
- How to solve the given problem of simple interest?
- Does $\displaystyle\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x) dx\,$?
- If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.
- **Ended Competition:** What is the shortest proof of $\exists x \forall y (D(x) \to D(y)) $?
- pointwise convergence and boundedness in norm imply weak convergence