All Questions
Tagged with conjectures prime-numbers
19 questions with no upvoted or accepted answers
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
5
votes
0
answers
326
views
Counting primes, twin primes, cousin primes: unusual approach, connection to some conjectures
I am investigating the following sieve-like algorithm. Let $S_N=\{1,\dots,N\}$. For all primes $p$ with $p_0\leq p \leq M$, we remove from $S_N$ the following elements: all numbers $n\in S_N$ such ...
- 3,098
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
748
views
Second Hardy-Littlewood Conjecture theme
If Second Hardy-Littlewood Conjecture is true then we can claim that $\pi(x)-\pi(y)\leq \pi(x-y)$. Thus the conjecture gives an upper bound for the number of primes between $x$ and $y$. I have found ...
user57432
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
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
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$$
$a(n)=0$ is $n$ being Wieferich number (not necessarily prime).
For odd $n$,...
- 25.4k
1
vote
0
answers
123
views
Testing polynomials irreducible over the integers
Let $f\in\operatorname{int}(\mathbb Z)$, the ring of integer-valued rational polynomials. Define $\operatorname{P}^+(f)$ as the number of primes $>0$ that $f$ assumes at distinct integral arguments....
- 862
1
vote
0
answers
142
views
Primes dividing functions defined by linear recurrence relations with constant coefficients
For Fibonacci numbers $F_n$ it holds that $p|F_{p-(\frac{5}{p})}$, if $p$ is an odd prime (Legendre symbol).
I guessed that the number $5$ came from the roots of the characteristic polynomial and ...
- 862
0
votes
0
answers
241
views
Conjecture about some recurrent primes
I want to know if there are conjectures similar to this one, I know there is the Bell primes conjecture or Gardner conjecture (mentioned in this page https://en.wikipedia.org/wiki/Bell_number), but ...
0
votes
0
answers
374
views
Is the Conjecture of Representing Integers as Differences of Semiprimes and Primes Extendable to Products of Distinct Primes?
Conjecture:
Let $k$ and $l$ be fixed distinct positive integers ($k≠l$). Then, for every positive integer $n$, there exist prime numbers $p_1,p_2,…,p_k∈\mathbb{P}$ and $q_1,q_2,…,q_l∈\mathbb{P}$ such ...
- 17
0
votes
0
answers
106
views
Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function
The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...
0
votes
0
answers
82
views
Inequalities $\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\pi(y^d)^\gamma$ involving the prime-counting function, where the constants are very close to $1$
Let $\pi(x)$ be the prime-counting function, I'm curious about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia)
$$\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\...