# 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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### Propagation of modularity and the Artin conjecture

The (still incomplete) solution of the Artin conjecture on dimension $\leq2$ has been a massive research effort that has spanned (knowingly or not) around a century. A very natural question is, what ...
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### Must the sum of the digits of $n^k$ decrease infinitely often, for $n,k\in\mathbb{N}$ and $n$ not a power of $10$?

This is a (rephrased) repost of a question I asked on MSE about 6 months ago, but didn't receive a definitive answer. Let $S(n)$ be the sum of the digits of $n$ (in base $10$), is it true that for ...
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### Fourier transforms and nontrivial vector bundles

We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of ...
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### Record analytic rank for an elliptic curve?

What is the current record (and reference) for the highest analytic rank of an elliptic curve over $\mathbb{Q}$? The highest algebraic rank is the Elkies curve with rank at least 28, but I cannot ...
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### Writing integers as a product of as few elements of $\{\frac21, \frac32, \frac43, \frac54, \ldots\}$ as possible

Is the number of elements we need to construct $x$ equal to $\log_2(x) + O(1)$? This question is inspired by question 2 of the 2018 European Girls' Mathematical Olympiad. I previously posted it on ...
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### Case D=4l in Elkies' paper on Supersingular Primes of an Elliptic Curve over $\mathbb{Q}$

My question is regarding Elkies' paper on "The existence of infinitely many supersingular primes for every elliptic curve over $\mathbb{Q}$". In the section "Nuts and Bolts", Elkies has the ...
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### Are the only solutions of the equation $\tau(n)$ mod $(n-1) = 1$ $n=4$ and $n=16$?

Define $\tau(k)$ as follows : $$\Delta(x)=x\prod_{n=1}^\infty(1-x^n)^{24}=\sum_{k=1}^\infty\tau(k)x^k.$$ This is equivalent to $$\prod_{n=1}^\infty(1-x^n)^{24}=\sum_{k=1}^\infty\tau(k)x^{k-1}.$$ ...
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### No Siegel-Landau zeros for $\mathrm{GL}(n)$

The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact: There ...
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### Between Fermat's primes and the twin primes

Let me start with a curiosity. The integers $11,13,17,19$ are prime numbers, and $101,103,107,109$ are prime as well. One might wonder whether there is another occurrence where $10^m+1,10^m+3,10^m+7$ ...