# 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

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### Third roots of unity and norm element

Let $K = \mathbb{Q}(\zeta_3)$ where $\zeta_3$ is a third root of unity, let $F = \mathbb{Q}(\sqrt[3]{\ell})$ where $\ell \equiv 1 \pmod{9}$ is a prime and set $L$ to be the Galois closure of $F$, i.e.,...
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### Small solutions of $x^2-a^3 y^2=\pm 1$

We are interested in small integer solutions to the Pell equation: $$x^2-a^3 y^2=\pm 1 \qquad (1)$$ Where in $\pm 1$ you can chose either sign. $(x^2,a^3 y^2)$ are consecutive powerful numbers. $abc$ ...
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### Cardinality of the set $\#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$

Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$. I am interested in upper bound for $$M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$$ where $N$ ...
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### heights of ideal classes and reduction theory for Bhargava cubes

Suppose $K$ is a quadratic imaginary field with discriminant $D$; let $S$ denote the ring of integers in $K$. For a fractional $S$-ideal $J$, define the height of $J$, denoted $H(J)$, to be the ...
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### Motivic $L$-functions came from automorphic representations

Langlands in his 1978 ICM talk made a conjecture that all motivic $L$-functions should arise as automorphic $L$-functions. A part of this conjecture, namely for some Hasse-Weil $\zeta$ functions is a ...
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### Fibonacci and matrix modular exponentiation

I'm interested in a few problems that are related enough that I decided to put them all in one question. What are the fastest known algorithms for finding large Fibonacci numbers modulo $p^k$, and ...
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### Limits related to the floor function

Here I am still interested in the function $f(n,k)=\frac{2^{k}+1}{2^{n}+1}\left\lfloor \frac{2^{n}+1 }{2^{k}+1}\right\rfloor$ and a Tauberian property that I would like to check. Let $\lambda>1$ be ...
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### Approximation of $\pi$ [closed]

I found a relation that gives $\pi$: $$\sum_{n=0}^{\infty} \frac{(2n)!}{2^{4n+1}(n!)^2(2n+1)}=\frac{\pi}{6}$$ To prove this formula, can we use the Maclaurin series? Thank you. EDIT: here is a second ...
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### Conditional convergence of Artin $L$-functions
Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition \$L(V,s)= \prod_v ...