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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 $n$,...
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Here is a new conjecture of mine from the appendix of an unpublished manuscript currently under review.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 ...
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 ...
5 votes
1 answer
811 views

A consequence of Firoozbakht's conjecture?

This is a question out of curiosity, while looking at the Firoozbakht's conjecture. It might not be research related, but as usual, I am not really sure if a question ever is research related or not, ...
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 $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$. ...
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 ...
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 ...
7 votes
1 answer
493 views

About semiprimal representations of $1$

Conjecture $A_1$ : For every $m \in \mathbb N \setminus \{1\}$ there exist mutually different primes $p_{r_1},...,p_{r_m}$ and mutually different primes $b_{w_1},...,b_{w_m}$ and numbers $i_1,...,i_m \...
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 $k$ ...
3 votes
1 answer
2k views

What do we know about Lucky numbers?

I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics. Wolfram states: write "out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The ...
2 votes
2 answers
484 views

On odd perfect numbers $p^k m^2$ with special prime $p$ satisfying $m^2 - p^k = 2^r t$ - Part II

(Preamble: We have asked this same question in MSE two weeks ago, without getting any answers. We have therefore cross-posted it to MO, hoping that it gets answered here.) The topic of odd perfect ...
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 > \...
3 votes
0 answers
328 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 ...
10 votes
1 answer
2k views

Is new $n$-conjecture as follows correct?

Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...
1 vote
1 answer
190 views

Infinitely many $k \in \mathbb{N}$ such that the closed interval $[k, k+99]$ contains from $2$ to $23$ prime numbers

Let $k \in \mathbb{Z}^+$. Is it possible to prove that, for some given $m \in \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23\}$, there are only finitely many $k$ such that the closed ...
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 ...
17 votes
5 answers
2k views

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?
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+\...
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 =...
2 votes
1 answer
307 views

Analogue of Fermat's little theorem for Bernoulli numbers

Is the following analogue of Fermat's Little Theorem for Bernoulli numbers true? Let $D_{2n}$ be the denominator of $\frac{B_{2n}}{4n}$ where $B_n$ is the $n$-th Bernoulli number. If $\gcd(a, D_{2n}) ...
20 votes
1 answer
594 views

Distinct exponents in the factorization of the factorial, a problem of Erdős

In the 1982 paper below, Paul Erdős proved that if $h(n)$ is the number of distinct exponents in the prime factorization of $n!$ then $$c_1\Big(\frac{n}{\log n}\Big)^{1/2} < h(n) < c_2\Big(\frac{...
7 votes
3 answers
1k views

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 ...
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-...
8 votes
2 answers
2k views

Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple. I have often heard of the statement that the SZC is stronger than the Riemann ...
5 votes
1 answer
403 views

Is there an even number $a$ such that $a^{2^{n}}+1$ is prime for infinitely many $n$?

Is there an even number $a$ such that $\{n: a^{2^{n}}+1 \text{ is prime} \}$ is an infinite set? Let $a$ be even. Is there infinitely many $n$ such that $a^{2^{n}}+1$ is composite?
-1 votes
1 answer
536 views

Questions about the abc conjecture [closed]

Question. Is there an integer $n_0 \geq 2$ such that $$\left\{\frac{c}{rad(abc)^{n_0}}: a, b >0,\; c=a+b,\; \gcd(a, b)=1\right \}$$ is bounded? The abc conjecture can directly deduce this ...
-4 votes
1 answer
882 views

Are there infinitely many $n$ such that $n!-1$ and $n!+1$ are prime numbers? [closed]

Are there infinitely many $n$ such that $n!−1$ and $n!+1$ are prime numbers?
1 vote
1 answer
316 views

Conjecture about primes [closed]

Let $n_0$ be an integer (positive or negative). Are there infinitely many primes $p$ such that $p + n_0 = {2^r} · q,$ $ r ≥ 0,$ $ q ≥ 3 $ is prime? When $n_0 = 2$, this conjecture is the twin prime ...
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 $S$ ...
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. ...
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, ...
2 votes
2 answers
642 views

On the nearest-square function and the quantity $m^2 - p^k$ where $p^k m^2$ is an odd perfect number

This question has been cross-posted from this MSE question and is an offshoot of this other MSE question. (Note that MSE user mathlove has posted an answer in MSE, which I could not completely ...
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 $\,...
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 ...
9 votes
1 answer
858 views

Prove that there are no composite integers $n=am+1$ such that $m \ | \ \phi(n)$

Let $n=am+1$ where $a $ and $m>1$ are positive integers and let $p$ be the least prime divisor of $m$. Prove that if $a<p$ and $ m \ | \ \phi(n)$ then $n$ is prime. This question is a ...
10 votes
3 answers
2k views

Have new conjectures generated by the Ramanujan machine been proven?

Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...
6 votes
1 answer
242 views

Inductively computing Mersenne primes / perfect numbers?

For two sets $A,B$ set $A+B = \{a +b | a \in A,b \in B\}$. Let $(x_n)_{n \in \mathbb{N}}$ be independent variables. Let $\sigma(n)$ be the sum of divisors of $n$. Set $\hat{\phi}(1) = \{x_1\}$ and ...
9 votes
1 answer
388 views

$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?

(I posted this question on Math SE but it has had no answer for a year now so I would like to ask if anyone here can provide one.) Thinking about the prime number theorem, I wondered whether it is ...
4 votes
1 answer
404 views

The action of the unitary divisors group on the set of divisors and odd perfect numbers

Let $n$ be a natural number. Let $U_n = \{d \in \mathbb{N}\mid d\mid n \text{ and } \gcd(d,n/d)=1 \}$ be the set of unitary divisors, $D_n$ be the set of divisors and $S_n=\{d \in \mathbb{N}\mid d^2 \...
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 $\...
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 ...
1 vote
1 answer
265 views

"Halfway" approach to Landau's 4th problem (special case of Bateman-Horn)

Landau's 4th problem asks if $n^2 + 1$ is prime for infinitely many $n \in \Bbb{Z}$. It is known that $n^2 + 1$ can only be divisible by Pythagorean primes, and that for any $p$ congruent to $1 \pmod ...
10 votes
1 answer
454 views

A constant bizarrely related to the Fibonacci Numbers

For roughly the past month, I have been studying denesting radicals. For example: the expression $\sqrt[3]{\sqrt[3]2-1}$ is a radical expression that contains another radical expression, so this ...
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 ...
9 votes
1 answer
418 views

Conjectured primality test for specific class of $N=4kp^n+1$

Can you provide a proof or counterexample for the following claim? Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $...
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 ...
0 votes
1 answer
780 views

Has Pillai conjecture been proven?

I found the paper https://hal.archives-ouvertes.fr/hal-00698687v9/document which claims the proof of Pillai conjecture. However, it is not mentioned anywhere that it has been proved. It's stated ...
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 \...
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 $\ \...