# 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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### Why did the Egyptians not represent $2/3$ as a sum of unit fractions in the Rhind papyrus?

The following is taken verbatim from the MathWorld Wolfram page on Egyptian fractions: An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated ...
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### Enrique Santos L's “Proof that no odd perfect number exists”

Background Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. A number $l$ is called perfect if $\sigma(l)=2l$. Let $n$ be an odd perfect number given in the so-called Eulerian ...
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### Improvement on the concept of separating families for the union-closed sets conjecture?

The union-closed sets conjecture states the following. Let $F$ be a finite family of finite sets that is union-closed and let $\cup (F)$ be the union of all sets in $F$. Then we can find an element in ...
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### Possible relationship between non-divisors of odd perfect numbers and coefficients of corresponding cyclotomic polynomials?

A positive integer $n$ is called perfect if $\sigma(n)=2n$, where $$\sigma(n)=\sum_{d \mid n}{d}$$ is the sum of divisors of $n$. If $n$ is odd and $\sigma(n)=2n$, then $n$ is called an odd perfect ...
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### On expressing $\frac{\pi^n}{4\cdot 3^{n-1}}$ as a continued fraction.

It is a celebrated equation that $$\frac{\pi}{4}=\cfrac{1}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\ddots}}}}$$ However, there are two other conjectured equations that I found which, if true (...
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### Minimal counterexample to $\frac{p - 1}{p^2}$-conjecture

There existed once a “folklore” conjecture that stated: Suppose $p$ is a prime. Then any finite group $G$ with $> (1 - \frac{p-1}{p^2})|G|$ elements of order $p$ has exponent $p$ This ...
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### proof of prime in every interval $(p^2,p^2+p)$

Overview We'll introduce a sort of little hack called the missing modulo conjecture which can identify an integer's previous prime. We then show that this may not be perfectly reliable on account of ...
### The Rook Conjecture: arrangement of $p$ primes being distinct $\pmod{p}$ through $p^2$
For any prime $p$, divide $[1,p^2]$ into $p$ equal intervals of length $p$, so that the first interval is $[1,p]$, the next $[p+1,2p]$, and so on. It is definitely unproven but seems likely that there ...