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Results tagged with nt.number-theory
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user 23571
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
53
votes
6
answers
9k
views
On Euler's polynomial $x^2+x+41$
This is an elementary question about something way outside my area of expertise. A well-known observation due to Euler is that the polynomial $P(x)=x^2+x+41$ takes on only prime values for the first …
- 1,389
3
votes
A cubic equation, and integers of the form $a^2+192b^2$
Some partial information from hand calculation:
The class number for discriminant $-768$ is 8 with the reduced forms $[1,0,192]$, $[3,0,64]$, $[4,4,49]$, $[7,\pm4,28]$, $[13,\pm8,16]$, and $[12,12,19] …
- 20k
5
votes
Integral solutions of quadratic equation $5 X² − 14 XY + 5 Y² = n$
Conway's construction of the topograph of a quadratic form $ax^2 + bxy +cy^2$ gives a way to find all integer solutions of an equation $ax^2 + bxy +cy^2=n$ since the topograph displays all the solutio …
- 20k
3
votes
Representation of integers by principal binary quadratic forms
As noted in an earlier answer referring to Pall's 1969 paper in the Journal of Number Theory, it is unlikely that there is a simple way to describe all the divisors of $k$ that are represented by the …
- 20k
6
votes
Periods of the continued fraction expansions of Galois-conjugate quadratic-irrationals
The periodic part of the continued fraction for the Galois conjugate is always the mirror image of the periodic part for the original quadratic irrational. Here we are viewing the periodic part as a c …
- 20k
4
votes
Accepted
Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion.
The structure of the automorphism group becomes clear when one looks at the Conway topograph of a given form. For an indefinite form not representing 0 the topograph has an infinite periodic river se …
- 20k
23
votes
Accepted
how to visualize the class number of an imaginary quadratic field?
This is an interesting question that I've wondered about myself, so I can't really answer it properly but I'll make a couple elementary observations. First, for a lattice in ${\mathbb Z}[\tau]\subset …
- 20k