Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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1answer
172 views

On the elliptic curve $x(x+a^2)(x+b^2) = y^2$

Ajai Choudhry showed that special cases of the elliptic curve, $$x(x+a^2)(x+b^2)=y^2\tag1$$ can be used to prove that, $$u_1^7+u_2^7+\dots + u_9^7 = 0\tag2$$ has an infinite number of primitive ...
0
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1answer
116 views

More on Vojta's exceptional set for a more general abc conjecture

Likely a mistake, but got very large exceptional set in Vojta's more general abc conjecture. In A more general abc conjecture, p. 7 Paul Vojta conjectures: If $x_0,\ldots x_{n-1}$ are nonzero ...
9
votes
0answers
155 views

Is the set of numbers $\{ [n^{3/2}] \mid n\text{ an integer}\}$ a basis of order 3?

A set of integers is called a basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of the set.(Nathanson, Additive Number Theory, page 7) ...
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0answers
74 views

Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
28
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0answers
520 views

Mod sequences that seem to become constant; and the number 316

Define a "mod sequence" of nonnegative integers based on one start parameter $s$, its first term, as follows. $A(s)=(a_1,a_2,\ldots,a_n,\ldots)$ with $a_1 = s$ and $$ a_n = \left(\sum_{k=1}^{n-1} a_k ...
2
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5answers
324 views

Mean of a vector

Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$ I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$. If I do it iteratively step ...
-6
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0answers
75 views

Can the divisibility-by-7 test be extended to preserve the remainder? [on hold]

The divisibility-by-7 test (in base 10) performs iterations whereby the last digit is doubled and subtracted from the number formed by all other digits. If the original number is divisible by 7, then ...
5
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1answer
464 views

Any way to prove Prime Number Theorem using Hyperbolic Geometry? [on hold]

The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$: $$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$ ...
1
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1answer
141 views

Prime divisors of $p^n+1$

Let $p$ be a rational prime and $n$ be a positive integer. It can be easily deduced from Zsigmondy's theorem that $p^n+1$ has a prime divisor greater than $2n$ except when $(p,n)=(2,3)$ or ...
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1answer
99 views

Algorithm Involving Quadratic Diophantine Equation

Let $a,b,c,d\in\Bbb Z$. $abcd\neq 0$ and $a>0$. What is the best standard low complexity procedure to find the set of solutions for $x,y,t\in\Bbb Z$ such that: $$-axy+bx+cy+d=t^2$$ where ...
20
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3answers
683 views

Number of elements of “$\mathrm{SL}_n(\mathbb{F}_p^\times)$” mod $p$

How many elements of $\mathrm{SL}_n(\mathbb{F}_p)$ have all nonzero entries? Just the answer mod $p$ would be fine as well. This seems like it should be easy/in the literature but I couldn't find it.
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0answers
84 views

$\zeta$ function for ambiguous class group

Let $K$ be a Galoisian number field with ring of integers $O_K$. I consider $\zeta_{A,K}(s)$ the function defined by $$\zeta_{A,K}(s)=\prod_{\substack{p\in\mathbb N\\p\text{ ...
5
votes
1answer
291 views

Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that their sum of prime factors is equal, counting repeated prime factors. (The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!) So ...
2
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1answer
144 views

Chebyshev polynomials factoring uniformly modulo all primes

Consider the Chebyshev polynomial of the first kind $T_n(x)$ and its factorization in $\mathbb F_p$ for a given prime $p$. Most often, this factorization is not uniform (meaning that the irreducible ...
7
votes
1answer
346 views

Tate's thesis for Artin L-functions

As far as I know, Tate's thesis has been successfully applied in two fronts: Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid) Automorphic L-functions, by Jacquet, Shalika, ...
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0answers
78 views

Relation between cyclotomic character and fundamental character of level

The question I have is the following: is it true that the $p+1$ exponentiation of the fundamental character of level $2$ gives us the reduction (mod $p$) of the cyclotomic character? For a review of ...
2
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3answers
250 views

Growth of $r_{2}(n)$

Let $n$ be a positive integer. From Jacobi's two-square theorem we know that the number $r_{2}(n)$ of representations of $n$ as a sum of two squares is given by $$ r_{2}(n)=4(d_{1}(n)-d_{3}(n)), $$ ...
6
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1answer
218 views

what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
10
votes
1answer
506 views

More elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?

(Note: See also the $a^4+b^4+c^4 = 1$ version in this old MSE post.) The equation discussed in a paper by Jacobi and Madden, $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 = z^4\tag1$$ or equivalently, $$(p-2q + ...
8
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1answer
548 views

What is a totient?

In addition to the Euler totient function, there are a great many generalizations and related functions which go by the "totient", usually with some name: Jordan, Lehmer*, Schemmel, Nagell, Alder, ...
10
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0answers
281 views

Applications of $p$-adic Hodge theory

I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. ...
1
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1answer
143 views

Bombieri-Vinogradov in short intervals

In 1985 Perelli, Pintz & Salerno proved a short-interval form of the Bombieri-Vinogradov theorem with $\theta \in (7/12, 1]$. Have there been any improvements on this, in particular with the ...
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0answers
53 views

The hypertriangular function of $n$

I'm looking for papers or recent results on the hypertriangular function of $n$: $$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$ This is A001923 in the OEIS. I don't have much experience with ...
3
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2answers
201 views

Sets of squares representing all squares up to $n^2$

Let $S_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$. Say that a subset $S \subseteq S_n$ square-represents $S_n^2$ if every square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting ...
2
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1answer
215 views
0
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0answers
66 views

Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?

If $\zeta(s)$ is the Riemann Zeta function, then $\zeta(n)^z$, with $z \in \mathbb{C}$, $\Re(s)>1$, can be represented as $$\zeta(s)^z=\sum_{n=1}^\infty \frac{d_z(n)}{n^{-s}}$$ where $d_z(n)$ ...
7
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0answers
362 views

Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following: Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...
1
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1answer
117 views

Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that $$ \left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ...
2
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3answers
730 views

Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...
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1answer
142 views

Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [closed]

Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that $$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$ ...
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0answers
45 views

Height of a tripod [closed]

I am trying to determine the height of a tripod when the length on the tripod's legs (81") and the distance between the ends that touch the ground are 57" apart. My thought process so far: When the ...
3
votes
1answer
136 views

Pathological behavior of Lie algebra under a map of abelian schemes

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...
5
votes
1answer
161 views

Which criteria for “uniformly splitting” polynomials?

Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...
5
votes
1answer
140 views

When is $\vartheta(x)>x$? [Skewes number analog]

Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$ Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing ...
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0answers
111 views

Shimura reciprocity law

I have spent quite some time understanding class fields generated by Kummer extensions and class fields generated by modular forms. Now, I am turning the notch of sophistication a bit to study class ...
1
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1answer
351 views

Numerical evidence and argument against Littlewood conjecture

This is joint work with someone. We got numerical evidence and argument against Littlewood conjecture, though mistakes are certainly possible. Littlewood conjecture states that for any two real ...
6
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1answer
250 views

abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of $$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$ obtained from the homotopy exact sequence of the étale ...
4
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1answer
427 views

Modular forms and “too many symmetries”

How do we interpret Barry Mazur's quote of Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence ...
2
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0answers
63 views

Varieties with few monomials and the n-conjecture

The n-conjecture is a generalization of abc and basically says that the if $a_1 + \ldots + a_n=0$, no proper subsum vanishes and $a_i$ are coprime, then the radical of $a_1\cdots a_n$ can't be too ...
7
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1answer
198 views

p-adic Stein spaces

The higher cohomology of coherent sheaves vanish on Stein spaces (both complex and p-adic). In the case when the space ($X$) is a curve and we're working in the complex world, this shows that all ...
2
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1answer
85 views

Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by, $$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$ with discriminant $D = (7 + ...
5
votes
0answers
511 views

Zeta function double product

Is it possible to write the following double product in terms of the zeta function? \begin{align} &\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}} \end{align} Extending the ...
7
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2answers
310 views

Least supersingular prime

Given an elliptic curve over the rationals, what can one say about the size of the smallest supersingular prime?
12
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4answers
581 views

Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as $$ \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ It is an entire function whose zeros are precisely those of $\zeta(s)$. Since $\xi$ is real ...
7
votes
2answers
304 views

Numbers with all N-digit prefixes divisible by N

In base 10, the number 3816547290 contains every digit exactly once. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. ...
0
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1answer
112 views

Short arithmetic progressions in quadratic residues

Let p be a prime number of form 4k+1. I guess that there are c(d) number of 3-terms arithmetic progressions (AP) in the set of quadratic residues modulo p, where c(d) is an integer constant depending ...
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2answers
164 views

Time estimate to determine if a number is prime [closed]

How long does it take to verify that a given number is a prime number, as a function of its number of digits, in a personal computer, say? How computationally hard is this?
11
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1answer
476 views

Is there a known primitive recursive upper bound on the nth “Zhang prime”

(This question is pure curiosity. Feel free to close it if you feel it is not appropriate for mathoverflow.) In 2013 Zhang showed that there are infinitely many pairs of primes which are less that ...
3
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1answer
220 views

“Weight-mondoromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...
2
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1answer
141 views

Special values of Hecke L-function

The Dedekind zeta function for a number field $K$ is defined as $\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$. By attaching a Hecke character $\psi$, we can define ...