All Questions
9 questions
3
votes
1
answer
203
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Chowla's theorem on class number of real quadratic field
Let $p\equiv1\bmod 4$ be a prime number and $h$
the class number of real quadratic field $\mathbb Q(\sqrt{p})$, $\epsilon=\frac{t+u\sqrt{p}}{2}$ its fundamental unit. In this paper https://www.pnas....
- 287
2
votes
0
answers
107
views
Record for determining complete list of imaginary quadratic fields with small class number
In 2003, Mark Watkins (Class numbers of imaginary quadratic fields) determined all imaginary quadratic fields having class number at most 100.
Has this list been improved? That is, what is the largest ...
- 26.9k
4
votes
1
answer
222
views
Generators of the ideal class group
Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following:
Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
- 43
0
votes
0
answers
64
views
What is the lattice point distribution over binary quadratic forms?
Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$.
For simplicity, we keep things only on quadrant I of the ...
-2
votes
2
answers
149
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Calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$ [closed]
How to calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$, where $n$ and $m$ are positive numbers.
We guess that: the great common factor is $1$.
- 577
1
vote
0
answers
84
views
How common are semiprimes with equally bitsized factors among semiprimes with equal bitsize?
I am curious about the following after having looked at the paper "Almost primes in almost all short intervals", theorem 3 says:
Almost all intervals $[x, x + \log^{3.51}{(x)}]$ with $x ≤ X$...
- 11
3
votes
1
answer
293
views
Number of lattice points on spheres with center not at the origin
Let $k\ge1$. It is known that the number of lattice points on the $k$-sphere $S^k(0)$ (center at the origin, radius $R$), namely the size of $\mathbb{Z}^{k+1}\cap S^k(0)$, is bounded by $R^{k-1+\...
- 187
0
votes
1
answer
159
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Parity and number of squares taken by polynomials in a range?
I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...
- 13.9k
9
votes
2
answers
647
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On bounds for idoneal integer
What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...
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