Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

10,877 questions
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Conjectured count of monogenic rings of fixed rank

By a monogenic ring of rank $n$ we mean a ring $R = \mathbb{Z}[\theta]$, where $\theta$ is an algebraic integer of degree $n$ over $\mathbb{Q}$. Put $f_\theta(x)$ for the minimal polynomial of $\theta$...
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Which Shimura varieties admit or don't admit $p$-adic uniformization by Drinfeld spaces?

$p$-adic uniformization is a powerful tool for studying Shimura curves and Shimura varieties. For instance, we know cohomology groups of Drinfeld spaces, so we know some information about the Shimura ...
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On a set of natural numbers

For any natural number $x$, let $f(x)$ be the natural number whose representation in base 3 is the same as the binary representation of $x$ (for instance, $f(5)=10$ because $5=101_2$ and $101_3=10$). ...
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Factorizations as a product of primes minus one

Let $x$ be a positive rational number. I am interested in factorizing $x$ as a product of primes minus one. In fact, I would also like make sure the primes in the decomposition are distinct, and I ...
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Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM

Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Grössencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$...
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Are the logarithms of the integer polynomials discrete in $L^1$ of the unit circle?

Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set. ...
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Modifying Liouville Number Construction

Liouville numbers are transcendental numbers that can be well approximated by rational numbers. A number x is a Liouville number if for every natural number $n$ there exist infinitely many pairs of ...
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Limit of the real part of a geometric sequence

I came across the following problem, which turned out to be surprisingly hard: Show that $\underset{n\rightarrow \infty}{\lim} \left| \mathrm{Re}((\frac{1+i\sqrt{7}}{2})^n)\right| = \infty.$ ...
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Extending prime numbers digit by digit while retaining primality [closed]

I looked at a table of primes and observed the following: If we choose $7$ can we concatenate one digit to the left so as to form a new prime number? Yes, concatenate $1$ to obtain $17$. Can we do ...
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Euler characteristics in the rank one case

Suppose $E$ is an elliptic curve over a number field with good ordinary reduction at the primes above a fixed odd prime $p$. We are interested in the Iwasawa theory over the cyclotomic $\mathbb{Z}_p$ ...
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Queries on distribution of prime divisors by magnitude?

Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$. What is the probability distribution of ...
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Magnitude and distribution of largest prime factor?

Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors. What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number? What ...