# Questions tagged [conjectures]

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

783 questions
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $2^n-n$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 - ...
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### 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 ...
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