# 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

**-4**

votes

**1**answer

43 views

### How to solve a Diophantine equation in six variables? [on hold]

Find all the integer solutions $a, b, c, d, e, f$ satisfying the equation $a^2b^3 + c^2d^3 = e^2f^3$.
Note that if we prove that there are no such solutions with the condition $\text{gcd}(ab, cd, ef) ...

**4**

votes

**1**answer

196 views

### Applications of the idea of deformation in algebraic geometry and other areas?

The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...

**2**

votes

**1**answer

249 views

### Are there infinitely many prime p, such that p=1296k^2+36k+7? [on hold]

I encountered a number theory problem when doing my research:
1.I want to know whether or not there are infinitely many primes $p$ satistying $gcd(\frac{p-1}{6},6)=1$, such that $6$ is a cubic ...

**1**

vote

**0**answers

51 views

### theta function with a low bound in the sum

I encounter a sum similar to the Jacobi Theta function except there is a lower bound $-J$ with $J\geq 0$:
\begin{equation}
f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n.
\end{equation}
My question is whether ...

**4**

votes

**1**answer

552 views

### Partial sums of primes

$2+3+5+7+11+13...$ is clearly the sum of the primes.
Now I consider partial sums such:
$2+3+5+7+11=28$ which is divisible by $7$
My question is:
are there infinitely many partial sums such that:
$...

**-3**

votes

**0**answers

164 views

### Fermat's last theorem related question [on hold]

From the $ABC$ Conjecture it is possible to derive the next result
$\bf{Proposition}$ $1.$ Consider some large enough different primes $p, q$, then the equation $A^pB^q + C^pD^q = E^pF^q$ has no ...

**0**

votes

**1**answer

112 views

### Calculating the number of solutions of integer linear equations

Let $N$ be a natural number. Consider the following set of matrices whose entries are non-negative integers:
$$X_N:=\left\{(c_{ij})_{i,j=1}^4\in M_4(\mathbb{Z}_{\geq 0})\bigg| \sum_j c_{1j} = \sum_i ...

**2**

votes

**0**answers

81 views

### Rationality of motivic zeta function for smooth Fano varieties

Assume the base field is complex number, and the coefficent is the Grothendieck ring of varieties (or it's localization at $[\mathbb L]$). We know the motivic zeta function for smooth projective ...

**2**

votes

**0**answers

54 views

### How to express the cuspidal form in terms of Poincare series?

Sorry to disturb. I recently read Blomer's paper. There is a blur which needs the expert's help here. Blomer's paper says "For any cusp form $f$ of integral weight $k$, level $N$, $f$ can be written ...

**0**

votes

**0**answers

107 views

### How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

Consider a system of $n$ divisibility conditions on $n$ prime variables:
$$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$
where $a_{i,j}$ are bounded integers. How many solutions ...

**1**

vote

**1**answer

104 views

### Power series rings and the formal generic fibre

Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements
\begin{equation*}
f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]
\end{equation*}
and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...

**2**

votes

**0**answers

86 views

### Satake correspondence for groups over finite field

I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.
In Langlands' program, Satake correspondence gives a correspondence between unramified ...

**3**

votes

**0**answers

51 views

### Supremum of certain modified zeta functions at 1

Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by
$$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$
where $(D/m)$ denotes the Jacobi symbol. ...

**5**

votes

**1**answer

113 views

### Modified Pascal's triangle

I posted this question to Mathematics Stack Exchange but got no answers. I hope that this question is advanced enough for this forum:
In Pascal's triangle, each number is the sum of the two numbers ...

**0**

votes

**0**answers

82 views

### Upper bounds for $|\theta(x)-x|$ assuming Riemann Hypothesis

What are the best currently known upper bounds for $|\theta(x)-x|$ assuming the Riemann Hypothesis, where $\theta(x)$ is the Chebyshev theta, and can someone provide the reference for this (not ...