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3 votes
1 answer
203 views

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....
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 ...
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 ...
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 views

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 $1$.
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 $x ≤ X$...
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+\...
0 votes
1 answer
159 views

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. ...
9 votes
2 answers
647 views

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 ...