# Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

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### Number of Solutions of the Hyperbola Equations over Finite Fields

I have a problem with proving the number of points of the hyperbola equation $H_a: x^2 + y^2 = a$ (for every a > 0 in finite field $F_p$) in the finite fields. I have to prove that the number of ...
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### Does there exists positive integers $m,n$ with $m\leq n$ and ${}_{n}m^k$ contains all of the $n$ digits for all sufficiently large integer $k$'s??

Let ${}_{n}x$ denote the base-$n$ ($n\geq 2$) representation of the integer $x$. Does there exists positive integers $m,n$ satisfying the following conditions $m\leq n$ $\exists k_0\gt 0$ such that ...
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### Silverman Proposition 2.5 computation

In the proof of Proposition 2.5 in Silverman's Arithmetic of Elliptic Curves, the author defines a map $$E_{ns} \to \overline{K}^*, \quad [X,Y,Z] \mapsto 1 + \frac{AX}{Y},$$ where $E_{ns}$ is the ...
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### Tate-Shafarevich group and number of points $\bmod p$

Is there any known relationships or conjectures between the size of the Tate-Shafarevich group and the number of points of the associated elliptic curves over $\mathbb{F}_p$. Thanks!
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### For every prime p there is a sum of squares congruent to -1 mod p [duplicate]

For every prime $p$, there exists $a,b \in \mathbb{Z}$ such that $p\mid a^2+b^2+1$ For context, this question shows up as a statement on a hint to showing that every positive integer is a sum of 4 ...
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### Mobius inversion formula proof [closed]

This is a part of the proof for mobius inversion formula I don't understand how do we get the second and third line. Please explain \begin{aligned} \sum_{d|n} \mu(d) F(n/d) &=& \sum_{d|n}\mu(d)...
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### How to prove this deduction in the Analytic Large Sieve using Beurling - Selberg function

I have been studying sieve theory from the following notes : http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html and I am struck on the following deduction in the proof of theorem 3.1 ( page 6) ...
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