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
17,513 questions
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Status of Mills' constant
There is a rather confusing state of affairs at Wikipedia concerning Mills' constant. The article on formula for primes mentions that It is not known whether it is irrational, but the article on Mills'...
-2
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0
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63
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Are 6n ± 1 or more general arithmetic progression forms used as a core tool in advanced Number Theory proofs, beyond basic sieving? [closed]
Is there any research that uses the 6n ± 1 form or the more general form kn ± r to prove more important prime number theorems? (e.g., linked to Dirichlet's theorem on arithmetic progressions)
3
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1
answer
244
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On sums of a prime and a central binomial coefficient
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
- 18k
-2
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Finding a family of solution to a diophantine problem
I have been working on this diophantine problem:
$Q^2 - ((6n+3)^2 + t) \cdot Q + 2(6n+3)(36n^3 + 54n^2 + 27n - 4) = 0$
For arbitrary values of $n$ from $n = 1,2,3,4,5,6,7,8,9,10,11,12,13,14$, the ...
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0
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123
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Is it possible that the Goldbach's conjecture or the twin prime conjecture be undecidable? [duplicate]
There are many famous unsolved problems in number theory that can be formulated by basic concepts. Two examples are
Goldbach's conjecture:
Every even natural number greater than 2 is the sum of two ...
3
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0
answers
139
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What do we know about $S(t+1)-S(t)$?
Let $S(t)$ denote $\frac{1}{\pi} \arg \zeta\left(\frac{1}{2} + i t\right)$, as usual. Do we have unconditional pointwise (i.e., not average) estimates on $S(t+1)-S(t)$ better than the ones we get from ...
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2
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124
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On the Peaks of Sum-Equals-Product Sequence at Prime Indices
The product of $n$ positive integers is equal to their sum, which can be expressed by the following equation：
$\prod_{i=1}^na_i=\sum_{i=1}^na_i$
For example, when $n=3$, we noticed that $(1,2,3)$ is ...
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1
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1
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150
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On even numbers of the form $p+p'+2^k$
Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that
$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$
...
- 18k
-1
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1
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Whether $2n>10$ can be written as $p+p'+2^a+2^b$ with $p$ and $p'$ consecutive primes?
In a paper published in 1971, R. Crocker proved that there are infinitely many positive odd numbers not of the form $p+2^a+2^b$ with $p$ prime and $a,b\in\mathbb Z^+=\{1,2,3,\ldots\}$. The proof makes ...
- 18k
0
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0
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105
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How to construct elliptic functions with predescribled zeros and poles by means of Weierstrass ℘-function and its derivatives?
Given a lattice $\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ in $\mathbb{C}$. It's well known that given $n_i\in\mathbb{Z}, z_i\in \mathbb{C}$ satisfying $\sum n_i z\in\Lambda$ and $\sum n_i=0$, ...
3
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0
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109
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Characterize nonzero integers via a polynomial in two variables
In a paper published in 1985, Shih-Ping Tung observed that an integer $m$ is nonzero if and only if $m=(2x+1)(3y+1)$ for some $x,y\in\mathbb Z$. In fact, we can write a nonzero integer $m$ as
$\pm3^a(...
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2
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180
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How many $n$-adic solutions of $y^{2 \cdot n-1} = y$ are there in $Z_n$?
I am studying fixed-point equations of the form $y^k = y$ ($k \in \mathbb{N}$) in $\mathbb{Z}_n$, where here $\mathbb{Z}_n$ denotes the ring of $n$-adic integers (i.e., the projective limit $\...
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1
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Exceptional catheti-lengths in primitive Pythagorean triples
when generating primitve Pythagorean triples and building sorted groups that contain a cathetus of specific length, I saw in one group an exceptionally high jump in the lengths of the second cathetus ...
- 14k
2
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130
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Can a holomorphic cusp form become a CM form？
Given a holomorphic cusp form $f$ with weight $k \geq 2$, level $N$ and trivial nebentypus. I am wondering if $f$ can be a CM (dihedral) cusp form, i.e. $f$ is isomorphic to its quadratic twist. ...
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8
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419
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Chudnovsky's work on Pillai's conjecture, mentioned by Guy
In the 2004 edition of Unsolved problems in Number Theory, Richard K. Guy asserted that Chudnovsky claimed to be able to prove that the gaps between perfect powers tend to infinity, essentialy proving ...
13
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342
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Dirichlet series that gives power of $\pi$ at positive even integer
Let $f(s)$ be a Dirichlet series with algebraic coefficients, and suppose that:
it admits a meromorphic continuation to $\mathbb{C}$;
there exists $d \in \mathbb{N}$ such that $f(2n) \in
\...
- 1,053
5
votes
1
answer
378
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Selberg's result on primes in short intervals
A result of Selberg (A. Selberg. On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid., 47(6):87–105, 1943) says essentially
$$\int ...
- 1,656
3
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1
answer
129
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Sum of prime divisors functions
I was idly thinking today about the functions $\displaystyle f(n) = \sum_{p \mid n} p$ and $\displaystyle F(n) = \sum_{p^e \| n} ep$, respectively the "sum of prime divisors" function and ...
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4
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1
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216
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Number field with rational real points
Consider a number field $K$. How to classify or characterize those $K$ intersecting the real line only in the rationals: $K\cap \mathbb{R}=\mathbb{Q}$ ?
An example are quadratic imaginary fields.
In ...
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1
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102
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Repulsion of integer points on elliptic curves
The question of bounding the number of integer points on an elliptic curve $E/\mathbb{Q}$, shown to always be finite by Siegel, is an old question. There are various aspects to this problem. The ...
- 30k
0
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0
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109
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Which of these two equivalent formulations is more natural to state an identity valid in every numeral system?
Let $r$ be an integer greater than $1$, and consider the radix-$r$ numeral
system. For any integer $a>1$ not divisible by $r$, we introduce the
base-$r$ congruence speed of the (integer) tetration ...
8
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2
answers
722
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Linear algebraic lemma in Weil II
The following linear algebraic lemma is used in Weil II to prove properties of $\tau$-real sheaves:
Lemma Let $A$ be a complex matrix and let $\overline A$ be its complex conjugate. Then the ...
- 4,732
8
votes
1
answer
365
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Coefficient comparison between a double sum and a single sum
For a formal Laurent series $F(q)$, denote its coefficient of $q^j$ by $[q^j](F)$.
QUESTION. For integers $r\geq1$, is this true?
$$[q^{2r}]\sum_{n\geq1}\frac{q^n}{1-q^{2n}}\sum_{k=1}^n\frac{q^k}{1+q^...
- 43.9k
2
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0
answers
183
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Are there bases admitting arbitrarily long runs of consecutive cute integers?
This is a followup to this question which in turn follows this post on Puzzling SE.
Say a positive integer is $b$-cute (or just cute if the base $b$ is clear in context) if it can be written as the ...
- 1,850
3
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133
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Alternative to the parity vector to describe the iterations in the $3\cdot x+1$ problem
As can be seen in the paper by Terras and Lagarias, it is possible to describe the results of the first $k$ iterations of the $3\cdot x+1$ problem by the parity vector $v(n)$.
If $s^{(i)}(n)$ are the ...
- 229
1
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1
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427
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A link between polynomial function and power
During my research, I came a cross this question :
Is it true that $\forall n \in \mathbb N^*,\exists P \in \mathbb Z[x],\forall k \in \mathbb N, 3^k \equiv P(k) \pmod{2^n}$ ?
- 6,007
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0
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42
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Linear-Disjointness of the field obtained upon iterated pre-images
Suppose $K$ is a number field and we have finitely many elements $\alpha_{1}, \ldots, \alpha_{k} \in K$ and a polynomial $f(x)\in K[x]$ of degree $\geq 2$. Given $m \geq 1$, we define $K_{m}(\alpha_{i}...
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180
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Identities of Bernoulli numbers from rank 2 simple Lie algebras
Denote $B_n$ as Bernoulli numbers, it is known that the following three identities hold (for $n\in \mathbb{N}$):
$$\tag{A}\label{504268_A}\frac{(2n)!}{(4n+1)!} \frac{-B_{6n+2}}{6n+2}= \sum_{k=0}^{2n} \...
- 1,053
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0
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117
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"Spreading out" the number of irreducible components
Let $\mathcal X$ be a scheme of finite type over $\mathbb Z$. We know that the number of geometrically irreducible components (of geometric fibers) is the constant at almost everywhere, by EGA. It is ...
- 415
0
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0
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217
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Finding new families of rational points to an elliptic curve problem
I have been working on this elliptic curve equation parameterized by $n \in \mathbb{Z}$:
$E_n : y^2 = x^3 + (-102n - 51) \cdot x - (432n^6 + 1296n^5 + 1620n^4 + 468n^3 - 513n^2 - 378n - 142)$
This ...
- 117
10
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1
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473
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Comparison between Faltings height and naive height of a curve
Let $C$ be a smooth curve of genus $g\ge 2$ over a number field $K$, and let $J$ be its Jacobian variety. Suppose we have an embedding $C\to \mathbb{P}^2$ so that $C$ is defined by a homogeneous ...
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Do all congruent numbers admit a rational point on $y^{2} = x^{6} + 4n^{2}x^{2}$with $y \pm 2nx$ both cubes?
$$y^{2} = x^{6} + 4n^{2}x^{2}$$
here $x,y$ have infinitely many rational solutions if and only if 'n' is a congruent number because this elliptic curve is a quadratic twist of the elliptic curve 32a1 ...
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4
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0
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288
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Congruence restrictions on prime divisors of polynomial values
Consider the polynomial $f(x)= x^2+1$. Can you prove that there are infinitely many integers $x$ such that $f(x)$ has no prime divisor congruent to $1 \bmod 3$? Obviously the prime divisors are ...
0
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0
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294
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Apéry series for $\zeta(3)$
I am still now stumped on deriving the series equivalence
$$\zeta(3)=\frac{5}{2}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$$
Like even I did not get the series for $\frac1{n^2}$ mentioned ...
- 2,135
4
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1
answer
356
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Ramanujan's work on the central factorial numbers
Background
The central factorial numbers are described on OEIS sequence A008955. Among the references, "Ramanujan's notebooks, part 1" (edited by Bruce Berndt) is listed. Upon checking this ...
- 5,065
2
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0
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224
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How cute can a number be, that is the product of two numbers sharing no digits?
Elsewhere (Numbers which are the product of two integers that share none of their digits) on this site cute numbers, numbers that can be expressed as the product of two numbers that share none of ...
- 3,659
3
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0
answers
127
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Burgess with an integer variable whose prime divisors are restricted?
Say a prime is a green prime if it is one of some given congruence classes modulo a fixed integer -- say, $p\equiv 1 \bmod 7$ or $p\equiv 5 \bmod 7$. Say an integer is green if all of its prime ...
- 22k
8
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1
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Do all primes $>2$ hit $5$?
$2$ is a fixed point of the iteration:
$$q_{n+1}:=\min_{p|(q_n-1)^2+1} p$$
Start with $q_1>2$ prime. Does this iteration hit $5$? (min runs over primes)
- 3,761
0
votes
1
answer
162
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Transcendental sum is also transcendental
If $a$ is transcendental, then is it necessarily true that $1^a+2^a+...+n^a$ is also transcendental for n>1?
- 1,223
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56
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Does a Motzkin-prefix greedy construction for distinct subset sums recover the Conway–Guy sequence?
A finite set $P=\{p_1>\dots>p_n\}\subset\mathbb{N}$ has distinct subset sums (DSS) if all $2^n$ subset sums are different. A problem of Erdős asks for $f(n)$, the minimal possible value of ...
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+50
Many degree $d$ nilpotent elements of quotients of polynomial rings and non-vanishing product
Generalization of this question.
Let $n$ be positive integer and $f_1(x_1,...,x_k), \dots, f_m(x_1,...x_k)$
polynomials with integer coefficients.
Let $K=\mathbb{Z}/n\mathbb{Z}[x_1,...,x_k]/\langle ...
- 25.7k
3
votes
3
answers
339
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Weakening of the Idoneal Number condition
This is about the following Math.SE Q&A: "What is the largest integer $d$ such there is a congruence relation on primes p so that $x^2 + dy^2 = p^2$ has a non-zero integral solution?".
...
- 26.6k
2
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1
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660
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A question about positive polynomials
Are there some $P,Q \in \mathbb R_+[x]$ with $(x+10)^{2025}=(x+2025)^2P(x)+(x+2024)^2Q(x)$ ?
PS : the AI give an negative answer in the case $(x+1)^{2025}$
I have posted the question here (*), but no ...
- 6,007
2
votes
1
answer
116
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Equidistribution for $k$ independent characters
Let $k$ be fixed. Let $q_1,q_2,\dotsc,q_k$ be coprime with product $\prod_{j=1}^k q_j$ of size about $N$. Let $n$ range over integers in $[1,\sqrt{N}]$ coprime to $q_1,q_2,\dotsc,q_k$. Is the vector $$...
- 22k
5
votes
0
answers
335
views
For a prime, is there always a prime number for which it is a primitive root?
Artin’s primitive root conjecture states that for any integer $a\neq \pm1$ which is not a square,there are infinitely many primes $p$ such that $a$ is a primitive root mod $p$. By Heath-Brown's result,...
- 528
4
votes
0
answers
272
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Minimum possible nonnegative value of $\pm a_1 \pm a_2 \pm \cdots \pm a_n$ for sufficiently large $n$
Conjecture. Let $M(n)$ denote the minimum possible nonnegative value of $\pm a_1 \pm a_2 \pm \cdots \pm a_n$ on all possible signs for a sequence of positive integers $a_n$ such that $\lim_{n \to \...
- 161
2
votes
0
answers
91
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Can $w^2+bx^2+cy^2+dz^2$ be universal over a sparse subset of $\mathbb N$?
Let $b,c,d\in\mathbb N=\{0,1,2,\ldots\}$ with $1\le b\le c\le d$. For a subset $S$ of $\mathbb N=\{0,1,2,\ldots\}$, if
$$\{w^2+bx^2+cy^2+dz^2:\ w,x,y,z\in S\}=\mathbb N$$
then we say that $w^2+bx^2+cy^...
- 18k
4
votes
1
answer
439
views
Is this strengthening of the Maynard-Tao theorem on primes in admissible tuples known?
The groundbreaking work of Maynard and Tao showed the following fundamental result:
For any integer $m$, there exists a number $k(m)$ such that for any admissible set $H$ of $k$ integers (with $k$ at ...
- 43
2
votes
1
answer
169
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Lower bounds for difference sets of finite integer sets
Let $S$ be a finite subset of integer. Let $\{p \leq X\}$ be the set of primes bounded by $X$. Is it true that the set $S-S$ has a subset $A$ of positive density such that
$p \mid a$ for all prime $p \...
- 317
7
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0
answers
297
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Is 1001 the only palindrome which is a product of three consecutive primes?
I made a computational search for over all integers $N < 10^{27}$.
Method:
Generate a list of primes up to $10^9$
Iterate over consecutive prime triples and compute the product
Check each product ...
- 115