# 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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### How can I catalog these generalized Collatz problems?

The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$: $x + x + 1 \rightarrow x+x+x+1+1;$ $x + x \rightarrow x;$ Whenever a number matches the LHS ...
1 vote
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### Imaginary quadratic fields with prime class number

Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$. In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write, "Since $h_K = p$, there ...
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### Anisotropic semisimple groups with no real compact factor

Let $F$ be a number field, and let $G$ be a semi-simple connected, anisotropic algebraic group over $F$ which is $F$-simple (or almost simple, the question is agnostic to isogenies). Suppose further ...
348 views

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### Prime splitting in the division field of an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
791 views

### Is every polynomial of the form $2x^{2n} -x^n +1$ irreducible over $\mathbb{Z}$?

Is every polynomial of the form $2x^{2n} - x^n +1$ irreducible for $n>0$? Motivation: A few years ago a student asked if $29$ was the largest number which is prime and one more than a perfect ...
1 vote
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### Iwahori action on the $p$-ordinary line of a principal series representation

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
57 views

### Upper bound for an additional Product formula

We have three sequences of positive integers $l$, $p$ and $q$ such that:  p_1 \geq p_2 \geq \cdots \geq p_k\text{ and } q_1 \geq q_2 \geq \cdots \geq q_k \geq \cdots \geq q_h \text{ where: } k < ...
1 vote
97 views

### If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$?

My present question is as is in the title: If $p^k m^2$ is an odd perfect number with special prime $p$, then must $m^2 - p^k = s^2 - t^2$ hold for some $s$ and $t$? It is known that $m^2 - p^k$ is ...
181 views

### Does RH emanate from a geometric interpretation of the Riemann $\xi$ function? [closed]

The Riemann $\xi$ function is defined in https://fr.m.wikipedia.org/wiki/Fonction_xi_de_Riemann. I realized that the critical line is the perpendicular bisector of the segment $[0,1]$, so that RH can ...
The Goldbach conjecure is not yet proved. But, when an even number is represented as a sum of two primes, is there any knwon result about the difference of the two primes? That is, if $2n$ is a sum of ...