# 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

12,328
questions

**11**

votes

**4**answers

1k views

### Mystery of the Monstrous Moonshine

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it,
...

**8**

votes

**1**answer

2k views

### The large sieve for primes

Let $\Lambda(n)$ be the von Mangoldt function, i.e., $\Lambda(n) = \log p$ for $n$ a prime power $p^k$ and $\Lambda(n) = 0$ for all $n$ that not prime powers. Let
$$S(\alpha) = \sum_{n \leq N} \...

**13**

votes

**1**answer

1k views

### What are the higher $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where $A$ is an abelian scheme?

Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian ...

**18**

votes

**4**answers

2k views

### Splitting Pythagorean triples

Can one partition the set of positive integers into finitely many Pythagorean-triple-free subsets? If so, what is the smallest number of such subsets? Taking a wild guess, I would
be least surprised ...

**19**

votes

**2**answers

3k views

### “Fermat's last theorem” and anabelian geometry?

Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes,...

**11**

votes

**6**answers

4k views

### Applications and Natural Occurrences of Prime Numbers

I'm fascinated by prime numbers, and over the years, I've found multiple applications and natural occurrences for them. But can anyone suggest some alternatives that aren't in my list?
Applications ...

**9**

votes

**3**answers

1k views

### Does Ribet's level lowering theorem hold for prime powers?

I often use the following theorem (that one can state more generally) in my research.
Let E/Q be an elliptic curve of conductor N corresponding to a modular form f(E), l a prime of good or ...

**13**

votes

**5**answers

2k views

### What is the Hilbert class field of a cyclotomic field?

In the answers to Qiaochu's post on defining representations of finite groups over the algebraic integers, it came out that which fields a representation of a finite group is defined over might depend ...

**-3**

votes

**2**answers

631 views

### Cycle Length of the Positive Powers of Two Mod Powers of Ten [closed]

I want to prove that the positive powers of two, mod 10m, cycle with period 4*5m-1. It's simple to prove that the powers of FIVE cycle with this period (2 is a primitive root mod powers of five), but ...

**69**

votes

**12**answers

8k views

### Is there a high-concept explanation for why characteristic 2 is special?

The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...

**14**

votes

**6**answers

2k views

### Why the search for ever larger primes?

I understand why primes are useful numbers and also why the product of large primes are useful such as for application in public key cryptography, but I am wondering why it is useful to continue the ...

**40**

votes

**1**answer

17k views

### What is inter-universal geometry?

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such ...

**15**

votes

**4**answers

2k views

### Arithmetic progressions without small primes

The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :
Is it known that there are infinitely many primes p for which ...

**8**

votes

**4**answers

1k views

### cohomology of moduli spaces

Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ ...

**18**

votes

**4**answers

2k views

### Exhibit an explicit bijection between irreducible polynomials over finite fields and Lyndon words.

Let $q$ be a power of a prime. It's well-known that the function $B(n, q) = \frac{1}{n} \sum_{d | n} \mu \left( \frac{n}{d} \right) q^d$ counts both the number of irreducible polynomials of degree $n$...

**2**

votes

**2**answers

935 views

### roots of analytic functions

Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $\mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic ...

**10**

votes

**1**answer

353 views

### An “existence contra partition of unity” statement for integer matrices?

While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...

**11**

votes

**1**answer

2k views

### Reference for the `standard' Tate curve argument.

I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...

**8**

votes

**2**answers

824 views

### Logarithmic structures on moduli of elliptic curves over Z

I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the ...

**58**

votes

**9**answers

17k views

### Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...

**31**

votes

**3**answers

3k views

### What is interesting/useful about big Witt Vectors?

$p$-typical Witt vectors are (among other things) a canonical way of associating to a perfect ring $A$ of characteristic $p$ a complete DVR of characteristic $0$ with residue ring $A$ generalizing $\...

**7**

votes

**4**answers

1k views

### Sums of cubes and more

It's well-known that every natural number can be written as a sum of 4 squares of integers.
Has there been any recent progress about the similar problem for the cubes, 4-th powers and so on? I ...

**8**

votes

**1**answer

818 views

### Generalized Teichmuller representatives

Fix a prime $p$. The Teichmuller representative associated to a $p$-adic integer $a$ is the unique root of $x^p - x$ in $Z_p$ congruent to $a$ mod $p$. One can identify this representative with the ...

**2**

votes

**3**answers

1k views

### What is the base change in number theory?

I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change ...

**0**

votes

**2**answers

3k views

### How to attack this diophantine equation in 3 variables?

From link:
Find integers a, b and c such that:
987654321a + 123456789b + c = (a + b +
c)³

**13**

votes

**5**answers

3k views

### Equivalent Statements of Riemann Hypothesis in the Weil Conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...

**10**

votes

**3**answers

910 views

### How to topologize X(R) when R is a topological ring?

Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...

**45**

votes

**5**answers

4k views

### Can $N^2$ have only digits 0 and 1, other than $N=10^k$?

Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's?
It seems very unlikely,...