# 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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### Possible connection between binary numbers and $\pi$

Here is the Desmos if you want to follow along: https://www.desmos.com/calculator/b4vtzruupm In messing around with binary numbers, I created a function $f(x)$ in Desmos that generated a list of ...
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### Question about the Erdős–Faber–Lovász conjecture

In graph theory, the Erdős–Faber–Lovász conjecture is a problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972. It says: If k complete graphs, ...
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### Prove this conjecture: Two lists of vectors are in the same orientation iff this transformation exists

Can you please help prove or disprove my conjecture below? Definition: Given vector space $V$ of $n$ dimensions, and ordered lists $x, y$ such that $x,y \subset V$; $x,y$ are each linearly independent;...
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### Collatz conjecture - Why does it end and not go on to infinity? [closed]

I was messing around with the sequences of odd numbers in the Collatz conjecture, and (unsurprisingly) found a pattern. Basically, I was calculating the number of steps an odd number takes to reach an ...
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### Conjecture: Shallow diagonals in Pascal's triangle form polynomials whose roots are all real, distinct, and in $(-2,2)$.

Here are the first few "shallow diagonals" in Pascal's triangle. We can use these shallow diagonals to make polynomials. Note that the even degree polynomials have an extra $\pm x$ at the ...
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### Conjecture about the minimum of the Gamma function

Problem/Conjecture: Let the function : $$f(x)=\frac{((x+x_{\min})!-(x_{\min})!)^{\frac{1}{x}}}{x^{\frac{1}{x^2}}}$$ Where $x_\min$ denotes the minimum abscissa of the Gamma function near by $0.4616$ ...
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### Conjectures with the strongest numerical evidence later proven to be false [duplicate]

For a number of conjectures in number theory, the truth of some statement has been checked up to extremely large values. E.g. There are no odd perfect numbers below $10^{1500}$ Are there any examples ...
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### Are there an infinity of consecutive primes that have no common digit?

Examples: $2$ & $3$, $3$ & $5$, $5$ & $7$, $59$ & $61$, $99999989$ & $100000007$. I was inspired by the fact that, in English, all consecutive numbers share a common letter, such ...
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### Revisiting MSE question 4386812 - Determining whether $\sigma(q^k)/2$ is squarefree, where $q^k n^2$ is an odd perfect number with special prime $q$

Preamble: The present inquiry is an offshoot of this earlier MSE question. MOTIVATION Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the ...
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### Some conjectures on additive and subtractive bases

I came up with a few problems about additive bases that might be interesting, would love to see if someone can prove or disprove them. Let S be a set of whole numbers so that every natural can be ...
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### (Almost) found a counterexample to the Beal conjecture [closed]

The Beal Conjecture states that if $A^x + B^y = C^z$, where $A, B, C, x, y, z$ are positive integers and $x$, $y$, $z$ are $≥ 3$, then $A$, $B$, and $C$ have a common prime factor. There is some ...
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### Do all Turing-complete systems converge on a single universal function w.r.t. description redundancy?

Let $L$ be any Turing-complete language. Let $d_L(n)$ be the number of distinct algorithms expressible in $L$ using at most $n$ bits. I'm not sure how to properly define "distinct algorithm",...
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### Graphs with all vertices having more neighbors than 2nd order neighbors

This is the conjecture that inspired my question: https://en.wikipedia.org/wiki/Second_neighborhood_problem For any vertex $v$, let $N(v)$ be the set of vertices adjacent to $v$ and let $N^2(v)$ the ...
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### What is the function E(x)?

I’m an electrical engineering student getting a minor in math. I recently started my first pure math class and my professor proposed something interesting on the board. I don’t remember exactly what ...
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### If $p^k m^2$ is an odd perfect number with special prime $p$ and $p = k$, then $\sigma(p^k)/2$ is not squarefree.

While researching the topic of odd perfect numbers, we came across the following implication, which we currently do not know how to prove: CONJECTURE: If $p^k m^2$ is an odd perfect number with ...
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### Given the $2^n$ primorial divisors, $D(5\#) = \{ 1, 2, 3, 5, 6, 10, 15, 30\}$, why are there always exactly $2^n$ different divisor indicator vectors?

Define $D(m)$ to be the (ordered) set of divisors of $m \in \Bbb{N}$. Let $m = p_n\# = 1 \cdot 2 \cdot 3 \cdot 5 \cdots p_n$. Clearly, $|D(m)| =$ the number of divisors of $p_n\# = 2^n$. Now define ...
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