All Questions
Tagged with conjectures nt.number-theory
30 questions with no upvoted or accepted answers
16
votes
0
answers
787
views
Numbers of the form $x^2(x-1) + y^2(y-1) + z^2(z-1)$ with $x,y,z\in\mathbb Z$
Roger Heath-Brown conjectured that any integer $k\not\equiv\pm4\pmod9$ can be expressed as $x^3+y^3+z^3$ with $x,y,z\in\mathbb Z$ in infinitely many different ways. Also it is well-known that some ...
- 701
15
votes
0
answers
365
views
Do primes of the form $4k+1$ ever lead the greatest prime factor race?
Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the ...
- 5,470
10
votes
0
answers
598
views
Does the interior of Pascal's triangle contain three consecutive integers?
This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
- 3,527
10
votes
0
answers
4k
views
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
0
answers
346
views
A generalization of Feit–Thompson conjecture, for square-free integers
I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...
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 $$\frac{\sigma(n)}n<e^\gamma \log\log n \...
- 13.4k
6
votes
0
answers
506
views
Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?
This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\...
5
votes
0
answers
192
views
Heuristics for the very little torsion in the cohomology of Shimura variety
Consider the following statement which is a part of Conjecture 1.3 in the paper titled "The asymptotic growth of torsion homology for arithmetic groups" authored by N. Bergeron and A. ...
- 443
5
votes
0
answers
614
views
is there a link with the probabilistic model for prime numbers?
Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$.
Let :
$$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
- 521
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
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
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
3
votes
0
answers
327
views
Conjecture about primes and Fibonacci numbers
I posted this conjecture on math.stackexchange, but I received no answer proving or disproving it: if $ m > 4 $ is a positive integer not divisible by $ 2 $ or $ 3 $, it's ever possible to find a ...
- 387
3
votes
0
answers
252
views
Counting twin primes with a sieve-like algorithm
The sequence A002822, denoted as $S$, represents all the twin primes except $\{3, 5\}$. Other than that exception, $k$ and $k+2$ are twin primes iff $(k+1)/6\in S$. Let $S(N)$ be the subset of $S$ ...
- 3,098
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 ...
2
votes
0
answers
113
views
On a subset of the $abc$ triples
The $abc$ conjecture states that, for every positive real $\varepsilon$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers such that $a + b = c$ and
$$c > \...
2
votes
0
answers
300
views
How soon can we represent a number as the sum of two primes?
Posting in MO since it was unanswered in MSE.
Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 =...
- 5,470
2
votes
0
answers
269
views
A relation of the prime counting function $\pi$ to counting the ordered ways of a number $n$ as a sum of two primes and two questions?
The definitions are from these two questions:
https://math.stackexchange.com/questions/3164216/a-series-related-to-prime-numbers
https://math.stackexchange.com/questions/4349186/trying-to-understand-...
- 3,064
2
votes
0
answers
114
views
A conjectured upper bound for the mean value of prime divisors inside prime gaps
In 1969 C.A. Grimm stated this interesting conjecture: the prime gap $\,G_n=\{x\in N:p_n\lt x\lt p_{n+1}\}\,$ contains at least $\,\#G_n=(p_{n+1}-p_n)-1=g_n-1\,$ distinct prime divisors, that is if $\,...
- 1,008
2
votes
0
answers
244
views
Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension
In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field.
If $K$ is an imaginary quadratic field and $F/K$ is ...
- 1,283
2
votes
0
answers
617
views
Arithmetic progression and average of two prime numbers
Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by:
$$
\ 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.
$$
For all terms of $A$ greater than $\ \...
- 359
1
vote
0
answers
77
views
Conjecture about Euler quotients related to non-Wieferich numbers $W(n)=\frac{2^n+1}{3}$
For odd natural $n$ define the Euler quotient:
$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$
$a(n)=0$ is $n$ being Wieferich number (not necessarily prime).
For odd $n$,...
- 25.4k
1
vote
0
answers
152
views
A new Conjecture at OEIS sequence A376842
Here is a new conjecture of mine from the appendix of an unpublished manuscript currently under review.
Let $b \in \mathbb{Z}^+$ and assume that $n$ is an integer greater than $1$ and not a multiple ...
- 1,451
1
vote
0
answers
158
views
Initial conditions to falsify Rowland's conjecture
Based on the Rowland's paper (A natural prime-generating recurrence), is there any theorem to show that for which initial condition $a(1) = k$ the conjecture can be falsified?
For example, for $k$ ...
- 151
1
vote
0
answers
47
views
On quasiperfect numbers, $\operatorname{rad}(\sigma(n))=\operatorname{rad}(2n+1)$ and $\operatorname{rad}(\sigma(n)-1)=\operatorname{rad}(2n)$
An integer $n\geq 1$ is said quasiperfect number if the sum of its positive divisors $\sigma(n)$ is equal to $2n+1$. See the Wikipedia Quasiperfect number.
The idea of this post is ask about the ...
1
vote
0
answers
222
views
Attempt of exploit the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$ in the context of even perfect numbers, and a related conjecture
It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...
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'...
0
votes
0
answers
241
views
Conjecture about some recurrent primes
I want to know if there are conjectures similar to this one, I know there is the Bell primes conjecture or Gardner conjecture (mentioned in this page https://en.wikipedia.org/wiki/Bell_number), but ...
0
votes
0
answers
374
views
Is the Conjecture of Representing Integers as Differences of Semiprimes and Primes Extendable to Products of Distinct Primes?
Conjecture:
Let $k$ and $l$ be fixed distinct positive integers ($k≠l$). Then, for every positive integer $n$, there exist prime numbers $p_1,p_2,…,p_k∈\mathbb{P}$ and $q_1,q_2,…,q_l∈\mathbb{P}$ such ...
- 17
0
votes
0
answers
82
views
Inequalities $\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\pi(y^d)^\gamma$ involving the prime-counting function, where the constants are very close to $1$
Let $\pi(x)$ be the prime-counting function, I'm curious about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia)
$$\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\...