All Questions
Tagged with conjectures nt.number-theory
96 questions
5
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1
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472
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Is the following weak version of second Hardy-Littlewood conjecture already known?
Very recently I was going through my previous MSE posts and I stumbled upon some of them regarding the Second Hardy-Littlewood Conjecture which states that,
For all $x,y\ge 2$ we have, $$\pi(x)+\...
user57432
3
votes
4
answers
1k
views
A conjecture regarding odd perfect numbers
(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of ...
8
votes
0
answers
257
views
Order of magnitude of extremely abundant numbers and RH
I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion (you can find the scanned paper here) that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \...
- 13.4k
48
votes
6
answers
5k
views
Are there examples of conjectures supported by heuristic arguments that have been finally disproved?
The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between ...
2
votes
0
answers
244
views
Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension
In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field.
If $K$ is an imaginary quadratic field and $F/K$ is ...
- 1,283
2
votes
2
answers
470
views
Is the exponential version of Catalan-Dickson conjecture true?
The aliquot sum function $s:\mathbb{N}\rightarrow \mathbb{N}$ assigns to any natural number $n$ the sum of its proper divisors. Perfect numbers are fixed points of this function. The open conjecture ...
10
votes
0
answers
4k
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Is the conjecture A+B=C following correct?
Is the conjecture on A+B=C following correct ?
Conjecture: Let $A, B, C$ be three positive integer numbers such that $A+B=C$ with $\gcd(A, B, C) = 1$. By Fundamental theorem of arithmetic we write:
$...
- 1,776
8
votes
3
answers
459
views
Density of the Klarner-Rado Sequence
Consider the Klarner-Rado sequence OEIS A005658 defined by the rule: the sequence starts with 1, and if it contains $n$ it also contains $2n$, $3n+2$ and $6n+3$. According to R. Guy's popular article,...
- 17k
7
votes
1
answer
230
views
Discriminant of numerator of inverse logarithmic derivative operator iteration
Let $T:\mathbb Q(x)\to \mathbb Q(x)$ be the operator of inverse logarithmic derivative, i.e. $$Tf=\frac{f}{f'}.$$ Define $$p_n(x)=T^n\left(x-\frac{x^2}{2}\right).$$ Let $f_n(x) \in \mathbb Z[x]$ be ...
- 7,193
3
votes
1
answer
842
views
Find all positive integers $n$ such that $n+\tau{(n)}=2\varphi{(n)}$
Conjecture：Today I have no intention of thinking about this question. I have only got two solutions so far. I guess there are only two solutions, but I won't prove it.
Let $n$ be positive integers, ...
- 4,280
2
votes
1
answer
14k
views
Are concatenations of two consecutive Mersenne numbers which are congruent to 6 mod 7 necessarily composite?
In this question on MSE, Enzo Creti asks for a prime number formed by concatenating the Mersenne numbers $2^n-1$ and $2^{n-1}-1$, for example, 40952047. For all residues modulo 7, he found primes ...
- 1,203
0
votes
1
answer
660
views
The difference between two coprime semiprimes
Conjecture:
Any positive integer can be written as the difference between two
coprime semiprimes.
Tested up to 1,000,000.
See also:
https://math.stackexchange.com/questions/2579578/the-...
- 862
62
votes
2
answers
3k
views
A conjecture regarding prime numbers
For $n,m \geq 3$, define $ P_n = \{ p : p$ is a prime such that $ p\leq n$ and $ p \nmid n \}$ .
For example :
$P_3= \{ 2 \}$
$P_4= \{ 3 \}$
$P_5= \{ 2, 3 \}$,
$P_6= \{ 5 \}$ and so on.
Claim: $...
- 653
18
votes
1
answer
2k
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Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$
Numerical evidence suggests a conjecture that the number of
points of certain elliptic curve over $\mathbb{F}_p$ is
either $p$ or $p+2$ for $p$ of certain form.
Let $p$ be prime of the form $p=27a^2+...
- 25.4k
5
votes
0
answers
238
views
The set of numbers $a+b$ such that $ma^2+nb^2$ is prime
Conjecture:
If $m,n$ are coprime it exist a minimal natural number $N_{mn}$ such
that:
$\{a+b>N_{mn}\mid a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P_{>2}\} = \{ k > N_{mn} \mid \...
- 862
3
votes
1
answer
355
views
Solve this diophantine equation: $m^4+n^4=10m^2n^2+1$
t's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo nn for every nn. This fact is stated, for example,
...
- 4,280
1
vote
1
answer
166
views
Question in the setting of generalized Diophantine $m$-tuples
As an amateur I am not quite sure should I post a question on the site for professional mathematicians but if the question is not appropriate for this site you can freely migrate it to ...
- 221
7
votes
2
answers
438
views
Generalization of Legendre`s conjecture
Legendre`s conjecture states that there is always a prime between $n^2$ and $(n+1)^2$ for every natural $n$.
It is natural to create following generalization:
Is it true that for every $\...
- 131
10
votes
1
answer
752
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A conjecture about certain values of the Fabius function
The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations
$$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$
and
$$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\...
- 6,819
26
votes
1
answer
1k
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What is the status on this conjecture on arithmetic progressions of primes?
The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes.
For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of ...
- 1,835
18
votes
3
answers
1k
views
A curious series related to the asymptotic behavior of the tetration
The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence
$$
{^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
- 6,819
0
votes
1
answer
161
views
Proving $k = 1 \implies q = 5$, if $q^k n^2$ is an odd perfect number with Euler prime $q$
Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$.
We want to show that the biconditional $k = 1 \iff q = 5$ holds.
It suffices to prove one direction, as the implication $q = 5 \...
4
votes
0
answers
317
views
there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$
Let $S_n:=1+\frac12+\frac13+\ldots+\frac1n$. Is it true that the set of $n\in\mathbb N$ such that
$$S_n-[S_n]<\dfrac{1}{n^2}$$
is infinite?
Here, $[x]$ represents the largest integer not exceeding $...
- 4,280
14
votes
2
answers
729
views
A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$
Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol:
$$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$
Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes ...
- 6,819
2
votes
0
answers
261
views
Any counter example for this: ${\phi(2^n-1)} \bmod \tau(2^n-1)=0$ for every integer $n \geq 1$? [closed]
I asked this question here In S.E but i don't received any resposnes for it, I would like to know if it is appropriate for M.O.
I'm always interesting for properties of the following series : $ \...
3
votes
0
answers
177
views
Looking for an appropriate reference(s) for two conjectures on odd perfect numbers
(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.)
Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For ...
1
vote
0
answers
119
views
If $N = q^k n^2$ is an odd perfect number, and $n < q^{k+1}$, does it follow that $k > 1$?
Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. According to Dickson (as pointed out recently by Beasley), Descartes conjectured $k=1$ in a letter to Mersenne in 1638, with Frenicle'...
2
votes
0
answers
204
views
On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers [closed]
(Note: This question has been cross-posted to MSE.)
Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$.
A number $M$ is called almost perfect if $\sigma(M) = 2M -...
1
vote
0
answers
301
views
Is this a proof of the Hardy-Littlewood inequality? [closed]
V.V. Miasoyedov posted a paper to the arXiv claiming a proof of the Hardy-Littlewood conjecture $\pi(x+y) \le \pi(x)+\pi(y)$. It seems a bit off, and not only because the conjecture is widely believed ...
- 9,114
24
votes
2
answers
2k
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A conjecture based on Wilson's theorem
Definitions:
Lagrange's theorem implies that for each prime $p$, the factors of $(p − 1)!$ can be arranged in unequal pairs, with the exception of $±1$, where the product of each pair $≡ 1 \pmod p$. ...
- 1,903
15
votes
2
answers
730
views
A conjecture about $\lfloor n!\cdot q/e\rfloor-\,!n\cdot q$
I was thinking about this question asked at Math.SE, when I came up with the following conjecture.
For every $q\in\mathbb Q$ consider a sequence $s_n^{(q)}$ (terms within the sequence are indexed by $...
- 6,819
5
votes
0
answers
425
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 $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$
...
- 161
1
vote
1
answer
362
views
A conjecture on the prime counting function
I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my earlier post I conjectured that the inequality ($\pi(x)$ denotes the ...
user57432
40
votes
2
answers
2k
views
Arctangents of odd powers of the golden ratio
While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of ...
- 6,819
15
votes
1
answer
1k
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What are the strongest conjectured uniform versions of Serre's Open Image Theorem?
This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...
- 1,499
3
votes
1
answer
222
views
If there are two primes at an even distance $k$ is there another disjoint pair of primes also at distance $k$?
Suppose $p,q$ are two primes at even distance $k$. Must there necessarily exist a different pair $p',q'$ composed of entirely different numbers such that $p'$ and $q'$ are also at distance $k$?
Edit: ...
- 1,835
2
votes
0
answers
617
views
Arithmetic progression and average of two prime numbers
Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by:
$$
\ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1.
$$
For all terms of $A$ greater than $\ \...
- 359
2
votes
1
answer
1k
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Need help publishing a mathematical proof? [closed]
Im not a mathematician with profession, but I know a group working on a Beal's Conjecture for years. They think that they have found a proof but the problem is that they don't know how to publish ...
1
vote
1
answer
338
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Representing the integers with powers of 2 and 3
Suppose that I have a number of the form
$$ x = \frac{1}{3^m}(2^{h} - \sum\limits_{k=1}^{m}3^{m-k}2^{v_k} ) $$
where m is a positive integer, and
$v_1 = 0 $
$v_{k+1} = v_k +1$ for $1 \leq k <...
- 119
5
votes
1
answer
605
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Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?
I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...
- 205
13
votes
2
answers
938
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On Generalizations of Fermat's Conjecture
We know the following facts:
(1) For all $1\leq n\leq 2$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}$ has a solution in $\mathbb{N}$.
(2) For all $3\leq n$ the equation $x_{1}^{n}+x_{2}^{n}=x_{3}^{n}...
user36136
12
votes
2
answers
370
views
A sequence based on Catalan–Mihăilescu problem
It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The conjecture was ...
- 532
11
votes
6
answers
3k
views
What are conjectures that are true for primes but then turned out to be false for some composite number?
Note: This is an update formulation since many people misunderstood the question before.
Of course it is easy to make a statement like "Every n is a prime or at most 1000", which is true for every ...
- 18.9k
7
votes
3
answers
1k
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Prime constellation conjectures
This is a simple question about terminology and provenance.
I just need to sort out the circle of conjectures that generalize and refine the twin prime conjecture.
I've encountered Polignac's ...
- 17.6k
8
votes
2
answers
3k
views
Is there any progress toward solving Gilbreath's conjecture?
Gilbreath's conjecture is a hypothesis, or a conjecture, in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the ...
- 479
17
votes
5
answers
2k
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Is there a progress on a solution of the inequality $\pi (m+n) \leq \pi (m) + \pi (n)$
in 1923 Hardy and Littlewood proposed the conjecture $\pi (m+n) \leq \pi (m) + \pi (n)$. Is there any progress towards solving this conjecture?
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