Timeline for Integers representable as binary quadratic forms
Current License: CC BY-SA 4.0
11 events
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| Mar 25, 2022 at 8:32 | history | edited | Bogdan Grechuk | CC BY-SA 4.0 |
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| Mar 25, 2022 at 4:06 | answer | added | Will Jagy | timeline score: 3 | |
| Mar 24, 2022 at 22:49 | comment | added | markvs | For example $x^2+11y^2$ has discriminant $44$. So if $n$ is odd prime and not $11$ then $n$ is represented by that form iff $\left(\frac{n}{11}\right)=1$. | |
| Mar 24, 2022 at 22:34 | comment | added | Bogdan Grechuk | @Gerry Myerson - thanks, I know that if two integers are representable in the form $x^2+11y^2$ then so is their product. But how this implies that the product cannot be also representable as $3x^2+2xy+4y^2$? After all, $15$ is representable as $15=2^2+11 (1)^2$ and also as $15=3(-1)^2+2(-1)(2)+4 (2)^2$. | |
| Mar 24, 2022 at 22:34 | comment | added | markvs | A prime number $n$ is represented by a binary form with discriminant $d$ iff $4d$ is a quadratic residue modulo $4n$ ($\gcd(n,d)=1$). | |
| Mar 24, 2022 at 22:17 | comment | added | Gerry Myerson | $(a+b\sqrt{-11})(c+d\sqrt{-11})=(ac-11bd)+(ad+bc)\sqrt{-11}$, so $(a^2+11b^2)(c^2+11d^2)=(ac-11bd)^2+11(ad+bc)^2$. | |
| Mar 24, 2022 at 19:47 | comment | added | Bogdan Grechuk | Thank you for the answer, but I thought that the principle class is the class of forms equivalent to the principal form. In this example, this is class of forms equivalent to $x^2+11y^2$, such as, for example, form $(x+y)^2+11y^2$, etc. When you write "primes are in the principle class", you probably mean some class of integers (not of forms), and this class is closed under the product operation. More details or reference would help. | |
| Mar 24, 2022 at 19:16 | comment | added | Stanley Yao Xiao | No. All prime of the form $x^2 + 11y^2$ are in the principal class, and any product of such primes are also in the principal class. | |
| Mar 24, 2022 at 19:07 | history | edited | Bogdan Grechuk | CC BY-SA 4.0 |
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| Mar 24, 2022 at 18:41 | history | edited | Bogdan Grechuk | CC BY-SA 4.0 |
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| Mar 24, 2022 at 18:33 | history | asked | Bogdan Grechuk | CC BY-SA 4.0 |