Questions tagged [conjectures]

For questions related to conjectures which are suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found

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7
votes
3answers
132 views

Conjecture: ${_2 F_1} \left(\frac{1}{2}- \frac{x}{2},1- \frac{x}{2};2;1 \right)=\prod_{n=2}^\infty \frac{(2n-2+x)(2n-1+x)}{2n (2n-3+2x)}$

Is the following conjecture for $0<x<1$ true and how do we prove it? $${_2 F_1} \left(\frac{1}{2}- \frac{x}{2},1- \frac{x}{2};2;1 \right)=\prod_{n=2}^\infty \frac{(2n-2+x)(2n-1+x)}{2n (2n-3+2x)}...
1
vote
1answer
211 views

Any positive integer can be written as sum / difference of consecutive squares

How should one go about proving that $x \in \mathbb{N}$ can be written (with the right combination of signs) as $\pm 1^2 \pm 2^2 \pm \ldots \pm n^2$ for any $x : x, n \in \mathbb N$? I have tried for ...
0
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1answer
80 views

Prime number function inequality conjecture

In my intuitive random search for conjectures I found $$n\neq 4\implies p_{n^2}\leq p_n^2<p_{4n^2}$$ It's tested for $n<1000$. I've looked at it the point of view of PNT but haven't the skills ...
7
votes
1answer
263 views

Recurrence relation for the Thue–Morse sequence

I made a curious observation. Let $a_n$ be the sequence of numbers determined by a recurrence relation $$\begin{cases} \vphantom{\large|}a_0=0\\ \vphantom{\large|}a_1=1\\ \vphantom{\Large|}n\,a_n=(5-...
4
votes
0answers
114 views

Number theoretical function with logarithmic properties

Introduction This is all about the function $v_a(x)$, what i could find about it and what questions i still have. First of all, I define $v_a(x)$ as the function that counts how often $x$ is divisible ...
7
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2answers
291 views

What methods show that a number is transcendental?

I've been doing a lot of research on such theories lately and these are all I've found so far: Liouvilles criterion (here) Lindemann-Weierstrass theorem (here) Gelfond-Schneider theorem (here) ...
4
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1answer
154 views

On Composite Numbers of the Form $p_{1}p_{2} \ldots p_{k} - 1$

This question is related to D. H. Lehmer's 1932 conjecture on Euler's totient function: Are there any composite $n$ for which $\phi(n)$ divides $n-1$? See, for example: On Lehmer's Totient ...
3
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0answers
174 views

Does there exist some prime $k$ for which there will be exactly two primes of the form $n!+k$?

This is a question related to my recent question Conjecture: “For every prime $k$ there will be at least one prime of the form $n!\pm k$” true? Using PARI/GP I searched for the number of primes of ...
18
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2answers
4k views

What do mathematicians mean when they say some conjecture can’t be proven using the current technology?

When reading about some open problems, a lot of them have quotes by renowned mathematicians that “[the conjecture] cannot be solved using the current technology” or something along these lines. What ...
14
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2answers
538 views

Conjecture: “For every prime $k$ there will be at least one prime of the form $n! \pm k$” true?

Using PARI/GP, I searched for primes of the form $n!\pm k$ where $k \ne 2$ is prime and $n\in \Bbb{N}$. With the help of user Peter, we covered a range of $k \le 10^7$ and couldn't find a prime $k$ ...
34
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2answers
990 views

Conjecture: Is $100$ the only square number of the form $a^b+b^a$?

Conjecture $100$ is the only square number of the form $a^b+b^a$ for integers $b>a>1$. In other words, $(a,b)=(2,6)$ is the only solution. Can we prove/disprove this? Observations The ...
1
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1answer
69 views

If a conjecture suggests $P(x)$ is true for all $x$ in an infinite domain, does that imply it is provable?

Let’s say a conjecture suggests for all $x$ in an infinite domain, $P(x)$ is true(conjectures such as the Goldbach Conjecture, the Collatz Conjecture, etc). Now, could we say no such conjecture could ...
12
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2answers
416 views

A conjectured value for $\operatorname{Re} \operatorname{Li}_4 (1 + i)$

In evaluating the integral given here it would seem that: $$\operatorname{Re} \operatorname{Li}_4 (1 + i) \stackrel{?}{=} -\frac{5}{16} \operatorname{Li}_4 \left (\frac{1}{2} \right ) + \frac{97}{...
10
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0answers
107 views

Conjecture: Is the identity $2^5-5^2=2+5$ unique? [duplicate]

Yet again a conjecture! Motivated by Catalan's conjecture and a recent question of mine, I conjecture that For distinct, positive integers $a,b$, the only solution to this equation $$a^b-b^a=a+b\...
0
votes
1answer
60 views

Searching for a conjecture that is true until the 127 power of n. [duplicate]

I am searching for a mathematical conjecture that is true until something like n = 117, or n = 127, or a number close. It is an ...
2
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0answers
69 views

A conjectured continued fraction involving non-polynomial patterns

After having been devoting some time for many years to experimental mathematics, I am thinking to publish the details of some of my most fruitful computing workflows for discovering identities. I ...
11
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0answers
158 views

Conjecture: No positive integer can be written as $x^y+y^x$ in more than one way

Today, I came up with the following problem when trying to solve this. Are there distinct integers $a,b,x,y>1$ such that the equation $$a^b+b^a=x^y+y^x$$ holds? That is, Is there ever an integer ...
0
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1answer
68 views

An arithmetic pattern

Define the odd part of a positive integer $\operatorname{od}(2^i(2j+1))=2j+1$ and define the function $f(n)=\operatorname{od}(n!-2^n)$. For $3\leq n \leq 10$ it holds that: $$f(2^n)-f(2^n-1)=2^n-2 \...
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0answers
25 views

Uniqueness of non-completable latin square of size $n$

It is known that partial latin squares of order $n$ and size $n-1$ can always be completed to a latin square. I want to know if non-completable partial latin squares of order $n$, size $n$ satisfy ...
7
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2answers
96 views

If $q$ is prime, can $\sigma(q^{k-1})$ and $\sigma(q^k)/2$ be both squares when $q \equiv 1 \pmod 4$ and $k \equiv 1 \pmod 4$?

This is related to this earlier MSE question. In particular, it appears that there is already a proof for the equivalence $$\sigma(q^{k-1}) \text{ is a square } \iff k = 1.$$ Let $\sigma(x)$ ...
77
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1answer
1k views

Conjectured formula for the Fabius function

The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions: a functional–integral equation$\require{action} \require{enclose}{^{\texttip{\dagger}{a ...
2
votes
1answer
61 views

Infinite $\frac{3}{2}$-generated groups?

Let’s call a group $\frac{3}{2}$-generated iff for every $g \in G \setminus \{1\}$ there is some $h \in G \setminus \langle g \rangle$ with $G = \langle g,h \rangle$. There is a conjecture by Breuer, ...
2
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0answers
36 views

Uniform residue limits over $\{1,2,\dots,p-1\}$

Let $p$ be a prime and $m \ge 1$. Define the ($p$ residue of a residue) function $\tag 1 \gamma_{(p,m)}: [1, pm] \to \{0,\dots,p-1\}$ $\tag 2 \quad \quad \quad \quad \quad \quad \quad \quad \quad \; ...
6
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0answers
141 views

Conjectures on the primality of $\sum\limits_{i=1}^n p_i^{p_i}$ and $\sum\limits_{i=1}^n (-1)^ip_i^{p_i}$

Inspired by Is $29$ the only prime of the form $p^p+2$?. Claims on prime powers and their alternating sums Consider the expressions $\mathcal P=\sum\limits_{i=1}^n p_i^{p_i}$ and $\mathcal Q=\sum\...
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1answer
138 views

Found a formula for generating all numbers in all possibilities of all cycle lengths

yo, I'm about to spread some new knowledge about the collatz conjecture. Not sure if this has been shown before or not, but here: https://en.wikipedia.org/wiki/Collatz_conjecture#Cycles it states that ...
1
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0answers
89 views

Refinement of Bertrand's Postulate

Bertrand's Postulate states that "for all $n \geq 2$, there is a prime $p$ between $n$ and $2n-2$". So a consequence is that, for all $n \geq 2$, there exists a prime $p \in (n, 2n)$. Let be $\delta: ...
12
votes
0answers
400 views

Is $\lfloor \zeta(-n) \rfloor$ only prime for $n=23$?

I searched for primes of the form of $\lfloor \zeta(-n) \rfloor$, where $n \in \Bbb{N}$, for a range of $n \le 10^4$ on PARI/GP and found $\lfloor \zeta(-n) \rfloor$ is only prime for $n=23$. My ...
12
votes
5answers
418 views

Prove that $\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$

Prove $$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$ Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my ...
2
votes
1answer
74 views

Does every simply-connected reductive group have trivial Galois cohomology?

Let $G$ be a linear algebraic group over a field $k$ with separable closure $k^s$ and absolute Galois group $\Gamma\!_k$. Consider the Galois cohomology group $$ H^1(\Gamma\!_k,G(k^s)). $$ This group ...
1
vote
1answer
140 views

Goldbach's conjecture among primes only

We couldn't so far prove that every even integer greater than 2 is the sum of two primes. Can something more be said about the following weaker form? Let $r,p,q$ denote primes $$\forall r\exists p,q\...
12
votes
1answer
181 views

An asymptotic series for $\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right),\,n\to\infty$

Using empirical methods, I conjectured that$^{[1]}$$\!^{[2]}$ $$\small\prod_{k=n}^\infty\operatorname{sinc}\left({2^{-k}}\pi\right)=1-\frac{2\pi^2}9\,4^{-n}+\frac{38 \,\pi ^4}{2025}\,4^{-2n}-\frac{...
1
vote
1answer
93 views

Rings for which the Köthe conjecture holds

Is there an overview of rings for which the Köthe conjecture is known to hold? In particular, I am interested in endomorphism rings of graded modules over multivariate polynomial rings. This survey ...
1
vote
2answers
130 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 ...
13
votes
0answers
343 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
vote
1answer
62 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
126 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 (...
2
votes
1answer
161 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
117 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 ...
8
votes
1answer
116 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
votes
2answers
123 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}...
6
votes
0answers
201 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 $2^n-n$...
54
votes
3answers
2k views

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
vote
0answers
71 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
votes
0answers
108 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
votes
0answers
16 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
votes
1answer
102 views

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 for ...
2
votes
1answer
71 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
votes
1answer
273 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...,\...
11
votes
2answers
192 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 ...
3
votes
1answer
80 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 ...

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