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Questions tagged [conjectures]

Use this tag if your question is about a well-known conjecture or a conjecture of your own.

0
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1answer
25 views

Collatz conjecture: $2^{m-1}(6n-3)$ is not part of any cycle

My original method was different from the method shown here. Instead of working my way backward through the iterations as below, I worked my way forward. I choose against doing that here despite of it ...
2
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0answers
26 views

Trigonometric integral related to Gieseking's constant

This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging ...
1
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1answer
55 views

A really nice and elementary conjecture involving numbers

Yesterday, i discovered a nice thing while playing with numbers. It is trivial to note that $\forall n\in \mathbb{Z^+},\exists x,y\in \mathbb{Z}$ such that $3^n=5x^2+y$ has solutions. Also, ...
1
vote
1answer
100 views

There cannot be more than three primes of the form $n^{n^2}-k$ for the same $k$?

I was searching for primes of the form $n^{n^2}-k$ on PARI/GP and noticed that primes of this form for same $k$ are quite rare. The probability of finding a prime of this form is $\frac {1}{n^2 \log (...
0
votes
0answers
167 views

Does no prime exist of the form of $k^k+11$? [on hold]

I tried searching for primes of the form $k^k+11$ on PARI/GP and found that no such prime exists for $k \le 10^4$. Questions: $(1)$ Is there any reason I cannot find a prime of the form $k^k+11$? ...
2
votes
1answer
108 views

Infinitude of super happy primes

Similar to happy primes, I define super happy primes by the following process: $(1)$ Find the sum of the digits raised to the power of themselves. Ex. $13$ gives sum $ = 1^1 + 3^3 = 28$ $(2)$ If ...
5
votes
1answer
62 views

Is there a maximum number of consecutive sexy prime pair sums that are all divisible by $10$?

I was finding the sums of pairs of sexy primes (prime numbers that differ by 6) and noticed that there are a lot of pairs who's sum is divisible by $10$. Ex. $(7, 13)$ as $7+13=20$ and $20$ is ...
2
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0answers
108 views

Is $6379$ the only prime $p \gt 2$ where $(p+1)!+1$ is prime? [closed]

I searched for primes of the form $(p+1)!+1$, where p is prime for a range of $2\lt p \le10^4$ on PARI/GP and found that $p=6379$ is the only prime in this range. Questions: $(1)$ Is $6379$ the ...
-1
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0answers
233 views

New primality test for $2m^n+1$ (where $m$ is prime)? [closed]

If $N=2.m^n+1$ (where $m$ is prime) you can prove if $N$ is prime or not by these two steps: Step (1) if $a^{2.m^{n-1}}=L \mod(N)$ (which is $L\neq1$ ) Step (2) $L^{m}=1 \mod(N)$ So N is prime. ...
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0answers
59 views

Is 29 the only prime of the form p^p+2 [duplicate]

searched for primes of the form p^p+2 but the only one I have found is 29
8
votes
1answer
91 views

Conjecture: all complex roots of $\sum_{k=0}^\infty \frac{z^k}{\left(nk\right)!}$ are real

Conjecture: $$\left[n\in\mathbb{Z}^+,z\in\mathbb{C},0=\sum_{k=0}^\infty \frac{z^k}{\left(nk\right)!}\right]\Rightarrow z\in\mathbb{R}$$ This conjecture has been verified for $n\in\{1,2,4\}$. The ...
0
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2answers
90 views

Will $p$ always be prime if $p^p+(p-1)!$ is prime?

While finding primes of the form $p^p+(p-1)!$ on PARI/GP, I noticed that $p$ is always prime if $p^p+(p-1)! \gt 2$ is prime. The search range was $p \le 10^5$. Here are the solutions for $p\in\Bbb{+Z}...
4
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0answers
167 views

Are $3$, $9$ the only natural numbers $n$ for which both $2^n-n$ and $2^n+n$ are prime?

I searched for natural numbers $n$, where $2^n-n$ and $2^n+n$ are both prime for the range of $n \le 10^5$ on PARI/GP and found that 3, 9 are the only solutions in this range. Note that since $2^n-n$...
41
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2answers
1k views
+50

Is $29$ the only prime of the form $p^p+2$?

I searched for primes of the form $p^p+2$, where $p$ is prime for a range of $p \le 10^5$ on PARI/GP and found that 29 is the only prime of this form in this range. Questions: $(1)$ Is $29$ the ...
1
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0answers
25 views

Create & Prove Conjecture - Discrete Math (Proofs)

I am stuck on the following problem: Imagine that a building has been overrun with snakes and rats. To help curb the problem, the building manager decides to offer employees brownie points for ...
3
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0answers
76 views

Circular Happy Palindromic Primes

$(1)$ A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. For example, 1193 is a ...
0
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0answers
9 views

Is showing all trees have $\rho$-valuation not enough to prove Ringel's conjecture about trees decomposing odd complete graph?

This might be a soft question, but I am trying to understand graceful labeling ($\beta$-valuation) and all the related stuff, and I have read Rosa's paper too. I would like to know why most are ...
3
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0answers
57 views
+50

Is it still unknown whetever any $\mathscr{D}$-class $D$ being a semigroup is bisimple?

Introduction: It is stated in my book from year 1961 that it is unknown whatever a subsemigroup $D$ of semigroup $S$, which is also a $\mathscr{D}$-class of $S$ is always bisimple, but it is known ...
2
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1answer
54 views

A question on discrete Fourier transform of some function

Let $\sigma(n) = \sum_{d|n} d$ and $\tau(n) = $ number of divisors of $n$. For each $k, 0 \le k \le n-1$ we can look at the discrete Fourier transform of the numbers $\sigma(\gcd(n,k))$ given by: $$\...
12
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1answer
213 views

Integrals of the Bessel function $J_0(x)$ over the intervals between its zeros

Let $J_0(x)$ be the Bessel function of the first kind. It has an infinite number of zeros on the positive real semi-axis. Let's denote them as $j_{0,n}$: $$j_{0,1}=2.40482...,\quad j_{0,2}=5.52007...,\...
9
votes
1answer
157 views

Asymptotic frequency of $[0,\,1,\,1]$ in the Thue–Morse sequence

Let $t_n$ be the Thue–Morse sequence: $$[\color{blue}{0,\,1,\,1,\,0,}\,\color{red}{1,\,0,\,0,\,1,}\,\color{blue}{1,\,0,\,0,\,1,}\,\color{red}{0,\,1,\,1,\,0,}\,...].\tag1$$ See this question for a ...
1
vote
1answer
60 views

A conjectured identity for the fractional iterates of $\sqrt{x}+2$

Deeply impressed by the Fractional iterates of $x²-2$, a problem by Ramanujan post one month ago, I decided to investigate a little by myself on the topic and discovered that using Carleman matrices ...
10
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2answers
248 views

Is the set of elementary functions which do not have elementary integrals bigger than set of elementary functions which have elementary integrals?

It increasingly seems to me that the functions that have elementary integrals are quite rare in comparison to the ones that don't have them. Even raising an elementary function to a different power ...
4
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0answers
46 views

Can the product of $n$ positive integers, where $n \gt 5$ in A.P. be a palindrome?

Reading the question can the product of four positive integers in A.P. be a square?, also made me question whether the product of $n$ positive integers, where $n \gt 5$ in arithmetic progression be a ...
0
votes
1answer
33 views

Proof that the error between $W(\lambda(\sqrt{n}))$ and the fraction of composite integers in $\mathbb{O}_n$ declines as $n$ increases.

Please provide assistance for the proof that the error between the function $W(\lambda(\sqrt{n}))$ and the fraction of composite integers in $\mathbb{O}_n$ declines as $n$ increases. Let $\mathbb{O}...
6
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2answers
271 views

Prove there is only one solution to the Diophantine equation $p^n - p = q^m - q$ where $p$ and $q$ are odd primes $p\gt q$

Consider numbers of the form $p^n - p$ where $p>2$ is a prime and $n>1 \in \mathbb{Z}$. How many of these have a unique representation? $2184$ can be written in this form $2$ ways, $3^7-3, 13^3-...
6
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0answers
68 views

New results on Identical Binomial Coefficient?

Are there any nontrivial identical binomial coefficients found other than: $$ {16 \choose 2}={10 \choose 3}=120 \\ {21 \choose 2}={10 \choose 4}=210 \\ {56 \choose 2}={22 \choose 3}=1540 \\ {120 \...
8
votes
1answer
172 views

Pairs of palindromic primes without $1$ and have a palindromic product

While discussing about prime numbers with other users, I noticed that: $(1)$ There are very few pairs of palindromic prime numbers that do not contain the digit $1$ and that have products which are ...
5
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0answers
78 views

Are 2, 3 the only prime numbers that don't have the digit 1 and are palindromes whose squares are also palindromes?

While thinking about prime numbers, I noticed that: $(1)$ Very few prime numbers have squares that are palindromes. Ex: $2$, $3$, $11$, $101$, $307$ $(2)$ Even rarer are prime numbers that are ...
8
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0answers
65 views

Are there infinitely many solutions such that the digit sum of a prime power is a smaller power of the same prime?

While discussing prime powers and divisors, I came up with the following problem. Examples $\to$ prime $p=3$ digit sum (in base ten) of $p^3=27$ is $p^2=9$, a power of $p$,. $\to$ prime $p=7$ ...
5
votes
1answer
89 views

Minimal order of a counterexample to Wall’s conjecture

There used to be once a rather well known and interesting conjecture, that was formulated by Gordon E. Wall: The number of maximal subgroups of a finite group $G$ does not exceed $|G|$ That ...
7
votes
1answer
81 views

For every $b$ in the power $a^{b}$, does there exist an $a$ such that the digit sum of this power is equal to $a$?

$1^0 = 1\to 1 =1$ $x^1=x\to x=x\;\forall x$. $9^2 = 81\to 8+1=9$ $8^3=512\to 5+1+2=8$. $7^4=2401\to 2+4+0+1=7$ $46^5 = 205962976\to 2+0+5+9+6+2+9+7+6=46$ $64^6 = 68719476736\to 6+8+7+1+9+4+7+6+7+...
5
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1answer
162 views

Is $\zeta(s)\sim\sqrt{\frac{\zeta(4s)}{\zeta(2s)}}\prod\limits_{n=1}^\infty\big(1-\frac{2}{p_n^s+p_n^{-s}}\big)^{-1/2}$?

The Riemann Zeta function, denoted by $\zeta(\cdot)$, is defined by the following equation for $s > 1$ and $p_n$ the $n^\text{th}$ prime number. $$\zeta(s)=\prod_{n=1}^\infty\bigg(1-\frac{1}{p_n^s}\...
2
votes
1answer
78 views

How does one subtract from concatenation in order to prove that $4\times 5 + 67 = 45 + 6\times 7$?

I noticed that if we get the numbers $4$, $5$, $6$ and $7$, they have an interesting property! $$4 \times 5 + 67 = 45 + 6 \times 7\tag*{= 87.}$$ I then conjectured that these were the only four ...
0
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0answers
16 views

Simple question on the polynomial variable in BMV trace conjecture

The solved BMV trace conjecture's equivalent states that if $A$ and $B$ are positive semidefinite matrices of order $n$, and $k$ is a positive integer, then the polynomial $p(t) = \text{Tr}\{(A+tB)^k\}...
2
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0answers
24 views

Conjecture on the growth of $q_1 = 1, q_{n+1}=q_n + f(q_n) $

This is a generalization of my answer to Calculate the limit of the following recurrent series in the form suggested by Will Jagy. $q_1 = 1, q_{n+1}=q_n + f(q_n) $ where $f(x) > 0$ and $f'(x) <...
1
vote
1answer
28 views

On a method to solve certain recursive sequences - looking for counterexamples?

When I started with this question, I wanted to know why my reasoning was wrong. Nevertheless, after checking some examples, I've noticed that my conjecture was actually - or at least seems to be - ...
1
vote
2answers
45 views

If $q$ is a Fermat prime, is $\sigma(q^k)/2$ a square if $k \equiv 1 \pmod 4$?

In what follows, let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number of the form $M = 2^{2^m} + 1$ is called a Fermat number. If in addition $M$ is prime, then $M$ is ...
1
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0answers
154 views

Constraints for the number of basis sets vs number of sets for a union-closed sets conjecture counterexample

I searched some literature about constraints on the number of basis sets (or $\cup$-irreducible sets, also called generators) $\vert\mathit{J}(\mathcal{F})\vert$ with respect to the number of sets $\...
1
vote
1answer
22 views

On a conjecture that two intersections are at $\big(\pm\frac{S}m, S\big)$ (about linear equations)

Suppose we have two linear equations, $y=mx$ and $y=-mx$ , inserted on the same plane for $m>0$. These equations are going to have an intersection at $(0,0)$. Now, let $$\begin{align}S&=1+\sum_{...
1
vote
1answer
51 views

Symmetry conditions for symmetric random vectors

While formulating the properties for a certain statistical model I'm dealing with, I came up with the following question (with credit going to MikeEarnest in comments for the proper formulation). A ...
0
votes
1answer
40 views

Sum on GCD and prime numbers

I was studying gcd then I encountered this sum $(1).$ A conjecture: If $(1)=1$ for any values of $N\ge3$, then N is a prime number. Let: $$f(N)=\frac{1}{N^{1-s}(N-1)}\sum_{j=1}^{N}(-1)^jj^s\frac{{...
5
votes
0answers
99 views

Conjecture about primes and the greatest common divisor

Conjecture: Given $m,n\in\mathbb N^+$, one odd and one even, there are two primes $p,q$ such that $|mp-nq|=\gcd(m,n)$. I hope MSE can determine its validity. From time to time, when testing my ...
3
votes
2answers
78 views

On a conjecture that $\sum_{i=1}\limits^na_ip_i = 1\Rightarrow \exists\big(\sum_{i=1}\limits^na_i=0\big)$ (about primes)

Conjecture: Denote by $p_i$ the $i^{\text{th}}$ prime number; by $a_i$ the $i^\text{th}$ arbitrary integer; and by $\exists(x)$ the existence of a chosen $x$.$$\sum_{i=1}^na_ip_i = 1\...
1
vote
2answers
84 views

An explicit “formula” for the prime counting function?

It is known that $\log(p_1),\cdots,\log(p_n)$ are linearly independet over $\mathbb{Q}$, where $p_i$ denotes the $i$-th prime. For a number $1 \le k \le n$ let $Log(k)$ denote the vector with respect ...
0
votes
0answers
24 views

Circumference of convex curves in the plane

For triangles, rectangles and ellipses in the plane the quote between the circumference and the diameter is invariant when the figures are magnified. The diameter is the maximum distance between ...
0
votes
1answer
83 views

Group (mathematics) Conjecture

Given $(G,•)$ as a Group with finite set $G$, operator •. Define: subset $S \subset G$ is called the core of the Group if and only if $$ \{ x•y ~|~ x \in S, y \in S \} = G \setminus S$$ ...
0
votes
1answer
56 views

A few questions regarding the function $f(x) = x+\exp(x)\cdot \log(x)$

The function $f(x) = x+\exp(x)\log(x)$ occurs prominently at Lagarias inequality: $\sigma(n) \le H_n + \exp(H_n)\log(H_n)$ where $\sigma(n)$ is the sum of divisors, and $H_n$ is the n-th harmonic ...
1
vote
2answers
53 views

Breaking the barriers at $q=5$ and $q=13$ for $q^k n^2$ an odd perfect number with special prime $q$

Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. If $\sigma(N)=2N$ and $N$ is odd, then $N$ is called an odd perfect number. The question of existence of odd perfect numbers is the ...
6
votes
0answers
140 views

Almost a prime number recurrence relation

For the number of partitions of n into prime parts $a(n)$ it holds $$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$ where $q(n)$ the sum of all different prime factors of $n$. Due to https://oeis....