**0**

votes

**1**answer

303 views

### Proof that $p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log {p_n}$ [closed]

I cannot find a proof of this theorem. May anyone assist?
$p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log{p_n}$

**8**

votes

**2**answers

782 views

### Update for 2015: least prime of form nq+1, with q prime?

I have received a complaint about my 2011 answer
least prime in a arithmetic progression
which, indeed, gives conflicting reports about this:
given a prime $q,$ what can we say about an upper ...

**4**

votes

**0**answers

73 views

### Smooth admissible representations, Hom, tensor and extension of scalars

(Remark: This has previously been posted on math.stackexchange, but I believe it might be suitable for this site as well. ...

**2**

votes

**2**answers

210 views

### The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$

I'm playing with exponential sums...
If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known
$$\sum_{x=0}^{q-1} e_q(ax^2) = ...

**6**

votes

**2**answers

424 views

### What is wrong with this deterministic algorithm efficiently generating large primes?

According to PolyMath
(Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in ...

**4**

votes

**1**answer

125 views

### Smoothness of Hecke algebras

First I will introduce some notation and definitions.
Fix a level $N$ (take $N=1$ if it makes things easier) and a prime $p$. Let $k$ be a finite field of characteristic $p$ and let $\mathcal{C}$ be ...

**7**

votes

**2**answers

309 views

### Theorems of the Galois groups of quintics appears not to work for the ${F}_{20}$ group determination

I am computing the Galois groups of quintics using the theorems from Ryan Kavanagh paper "On Irreducible Rational Quintics" using the decic resolvent ${P}_{10} \left({x}\right) = \prod\limits_{1 \le i ...

**2**

votes

**2**answers

247 views

### Asymptotics of the least common multiple of the first natural numbers

What is $$ \limsup_{n \to \infty} \frac{\log(\mathrm{lcm}(1,2, \dots, n))}{n} \ \ ?$$

**4**

votes

**1**answer

256 views

### Integer solution

For every prime $p$, does there exists integers $x_1$, $x_2$ and $x_3$ ($0\leq x_1, x_2, x_3 \leq \lfloor cp^{1/3}\rfloor$ and $c$ is some large constant) such that $\frac{p-1}{2}-\lfloor 2cp^{1/3} ...

**3**

votes

**1**answer

95 views

### Upper bound for OEIS A076689 “Smallest k such that k*p#+1 is prime”?

OEIS A076689
Is defined as smallest integer $a(n)=k$ such that $k n\#+1$ is prime,
where $n\#$ is primorial, the product of the first $n$ primes.
Lower bound appears $1$, the primorial primes.
...

**-3**

votes

**2**answers

287 views

### The number of totatives to the nth primorial, in an interval shorter than the nth primorial

(The notation of this question will be improved over the next few days, sorry for the lack of clarity at the moment.)
Can, and if so when can, we determine the amount of natural numbers which are ...

**2**

votes

**1**answer

123 views

### Which values of symmetric square $L$-functions are critical?

I've recently been learning about the special values of symmetric square $L$-functions of modular forms.
If $f$ is a cuspidal modular eigenform (of some weight $k \ge 2$) then its symmetric square ...

**9**

votes

**2**answers

549 views

### Many representations as a sum of three squares

Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...

**0**

votes

**1**answer

130 views

### Primes in simultaneous arithmetic progressions

Suppose we're given four positive integers $a$, $b$, $c$, $d$ such that $a$ and $b$ are coprime, and $c$ and $d$ are coprime. Is there a non-negative integer $k$ such that both $ak+b$ and $ck+d$ are ...

**3**

votes

**2**answers

145 views

### Well-distribution of square of an interval $[1,p^{1-\varepsilon}]$ modulo a prime $p$

Let $p$ be a large prime. I would like to say that the multi-set $[1,p^{1-\varepsilon}]^2 = \{ab \mod p: a, b \in [1,p^{1-\varepsilon}]\}$ is close to uniformly distributed, i.e. that every nonzero ...

**6**

votes

**1**answer

317 views

### Uniform definition of $S(\mathbb{R})$ and $S(\mathbb{Q}_p)$

Let $\mathcal{P}=\{\infty, 2,3,5,7,11,\ldots\}$ be the set of primes of $\mathbb{Q}$ and let $\mathbb{Q}_p$ denote the corresponding completions, so in particular $\mathbb{Q}_{\infty}=\mathbb{R}$.
Is ...

**12**

votes

**1**answer

230 views

### To what extent are modular parametrizations expected to generalize?

By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) ...

**5**

votes

**0**answers

143 views

**1**

vote

**0**answers

104 views

### On the Weil Chatelet Group

Let $A$ be a abelian curve over a number field $K$. The Weil Chatelet group parametrizes the twists of $A$, modulo the twists with a $K$ rational point. We can assume that $A$ is a plane curve. My ...

**14**

votes

**6**answers

1k views

### primorial puzzlement

Let $x_n$ be the smallest positive integer which is not a quadratic residue modulo any of the first $n$ odd primes. The question is: is there any bound on how quickly $x_n$ grows as a function of $n?$ ...

**6**

votes

**2**answers

495 views

### On bounds for idoneal integer

What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...

**0**

votes

**0**answers

36 views

### How much p-adic logarithm with precision 2 speeds discrete logarithm modulo $p^2$?

Let $p$ be large prime and $g$ generator of the multiplicative
group modulo $p^2$.
The discrete logarithm (DL) modulo $p^2$ asks : given $g,A,p$, find $x$
satisfying $g^x \equiv A \pmod{p^2}$.
...

**3**

votes

**2**answers

194 views

### Is the sumset or the sumset of the square set always large?

Let A be a finite subset of $\mathbb{N}$, $\mathbb{R}$, or a sufficiently small subset of $\mathbb{F}_{p}$.
Do we have a lower bound of the form $|A|^{1+\delta}$ on the following quantity:
$$\max ...

**7**

votes

**0**answers

164 views

### Are there an infinite number of twin semiprimes?

A semiprime is a number that is the
product of two (possibly equal) primes.
Define twin semiprimes (my terminology) as two consecutive numbers both semiprimes.
For example, $(57,58)$ are twin ...

**6**

votes

**0**answers

368 views

### Prime gap counts in short intervals

Since it is conjectured that the twin prime count at $n\sim2 C_2\ \frac{n}{\log^2n},$ where $C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} = 0.66016 18158 \dots,$ it follows that the twin prime count ...

**8**

votes

**2**answers

306 views

### Determining the Lambert series for $xq+x^2q^4+x^3q^9+…+x^nq^{n^2}+…$

I am trying to determine the polynomials $P_n(x)$ from
$$
xq+x^2q^4+x^3q^9+...+\ x^nq^{n^2}+...=\sum_{n\geqslant1}\frac{P_n(x)q^n}{1-xq^n};
$$
that is,
$$
\sum_{d|n}x^{\frac ...

**3**

votes

**1**answer

60 views

### Avoiding the range of a bivariate function or Diophantine function

I have a bivariate integer function where x,y are positive integers in the function $f(x,y)=5+23x+7y+30xy$. The lattice points of this function, or its range, contain a large number of values. I'm ...

**5**

votes

**1**answer

157 views

### Dynamics of the distribution of prime factorization types in increasing intervals

I've tagged this as reference request as surely this question must be very well investigated, I just don't know how to look for it. Most likely the perfect answer will be in form of a keyword for ...

**9**

votes

**0**answers

214 views

### Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?

If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property?
For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an ...

**23**

votes

**0**answers

500 views

### Relaxed Collatz 3x+1 conjecture

The Collatz $3x+1$ conjecture claims that any positive integer can be eventually reduced to 1 by iterative application of the maps $x\mapsto 3x+1$ whenever $x$ is odd and $x\mapsto x/2$ whenever $x$ ...

**2**

votes

**0**answers

100 views

### Effective Realization of GCD of middle binomials?

So, it is well-known that
$$ \gcd \left(\binom{m}{k}\mid 1\leq k\lt m \right) = e^{\Lambda(m)}$$
which can incidentally be sparsified for prime $p$
$$ \gcd ...

**1**

vote

**2**answers

106 views

### Asymptotic of the sum of squared primes [closed]

I have a rather simple question of number theory which I can't seem to be able to find a good reference for. I am not a specialist and I don't really know where to look. I would like to show that the ...

**2**

votes

**1**answer

99 views

### Mestre-type algorithm for higher-genus curves?

Is there an analogous algorithm for genus $g>2$ curves that, given a complete set of invariants, outputs a curve with those invariants?
(I'm interested in particular in $g=3$.)
Any references ...

**5**

votes

**0**answers

87 views

### Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...

**6**

votes

**1**answer

295 views

### Negative coefficient in an almost cyclotomic polynomial

Let $a,b,c,d$ be four prime numbers. We set the polynomial :
...

**2**

votes

**0**answers

49 views

### Questions about holomorphy and zeros of the symmetric power $L$-function

Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions ...

**11**

votes

**2**answers

735 views

### Can I find Fermat's complete works anywhere?

I admire the mathematician very much and want to look at his writings. Is there anywhere in book or web form that has a collection of his writings?

**3**

votes

**0**answers

181 views

### Probability that an integer contains no $1\bmod 4$ prime factor

$n$ represents integer variable.
What is the probability that and integer contains at most $r(n)$ prime factors of form $1\bmod 4$ where $r(n)$ is a function of $\omega(n)$ (number of distinct prime ...

**-1**

votes

**1**answer

118 views

### Counting number of primorial factors

Denote $$P(n)=\prod_{p\in\mathsf{Primes}\leq n}p$$ signifying $n^{\mbox{th}}$ primorial.
We know that $P(n)$ has approximately $n/\log2$ bits ...

**7**

votes

**1**answer

616 views

### A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...

**7**

votes

**1**answer

226 views

### Sums of reciprocals involving divisor sums

This question was asked at MSE but never received an answer.
Let $A\subset\mathbb{N}$ be a subset of the natural numbers, and let $\sigma(n)$
denote the sum of divisors of $n$. Recall that we have ...

**4**

votes

**0**answers

248 views

### Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ?
Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$
...

**1**

vote

**1**answer

107 views

### Need an explanation of a deduction

When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the ...

**13**

votes

**2**answers

378 views

### Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$.
Probable primes are the union of the primes and base two pseudoprimes.
This definition is much ...

**6**

votes

**1**answer

422 views

### Sums of unique squares

Let $\mathbb{N}$ denote the positive integers and let $S = \{n^2: n\in \mathbb{N}\}$. For any positive integer $k$ we define $$\text{sq}(k) = |\{F\subseteq S: F\neq \emptyset, F\text{ is finite and } ...

**2**

votes

**0**answers

64 views

### Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...

**14**

votes

**2**answers

447 views

### Number of distinct factors

Denote $\omega(m)$ to be number of distinct factors of $m$ as defined in http://mathworld.wolfram.com/DistinctPrimeFactors.html.
At every $c>0$, given $n\in\Bbb N$ define $$S(n,c)=\big\{m\in\Bbb ...

**6**

votes

**1**answer

272 views

### Irreducible cubics modulo primes

Is there a small finite (perhaps of cardinality two or three) collection of cubic polynomials $p_1, \dotsc, p_k \in \mathbb{Z}[x]$ such that for every prime $p$ at least one of these is irreducible?

**2**

votes

**1**answer

130 views

### Odds of residue being small

Given $\mathsf{c\geq1}$, what is the probability that if you choose $\mathsf{A,B,\alpha\in\Bbb N}$ such that $\mathsf{A,B<\alpha<AB}$ holds we will have both ...

**0**

votes

**1**answer

85 views

### Question about sign change of Hecke eigenvalues

I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper ...