60 problems

Dris's conjecture. The special-prime component is smaller than the remaining square-root component:

Sorli's conjecture. The exponent of the special prime is one:

Let $n$ be an odd perfect number in Eulerian form … where $p$ is prime, $p\equiv 1\pmod 4$, $k\geq 0$, and $c$ is a positive integer not divisible by $p$. Write … with distinct pri…

Let $N$ be an odd perfect number of Eulerian form … where $p,q_1,\ldots,q_{r-1}$ are distinct odd primes, $\beta_1,\ldots,\beta_{r-1}$ and $\alpha$ are positive integers, and…

Sylvester's conjecture. There is no odd perfect number.

Let $S$ be a fixed finite set of integers, let $n/d$ be a fixed rational number, and let $s$ be a fixed integer. An odd $n/d$-perfect number is an odd integer $N$ satisfying…

Let $n$ be a positive odd integer. Call $n$ abundant if $\sigma(n) \geq 2n$, and call it nonsquare if it is not a perfect square. Nonsquare odd abundant numbers conjecture. Every n…

Let $\mathcal{A}$ denote the density of the abundant positive integers, and let the density of the positive $\{-1,1\}$-perfect numbers mean the corresponding natural density when i…

Let $n$ be a positive integer. Call $n$ abundant if … ; a positive integer is {-1,1}-perfect when it has the property defined in the paper. Almost-all abundant numbers conjecture.…

Recursive divisor-sum sequence conjecture. For every positive integer $n$, the sequence $(x_k)$ eventually reaches $1$.

Odd perfect number conjecture. There are no odd perfect numbers.

Perfect-number conjecture. There are infinitely many even perfect numbers, but no odd perfect numbers.

For an integer $\mu$, a $\mu$-Sondow number is a positive integer $n$ such that … where the sum is over the prime divisors $p$ of $n$. A strongly pseudoperfect number is understood…

A weak primary pseudoperfect number is a positive integer $n$ satisfying … where the sum is over the prime divisors of $n$. A strongly pseudoperfect number is understood in the sen…

A primary pseudoperfect number is a positive integer $n$ satisfying … where the sum is over the prime divisors $p$ of $n$. A strongly pseudoperfect number is understood in the sens…

Let $x,y\in\mathbb{N}$ be coprime and satisfy $x\ne y$. Distinctness conjecture. The values … are distinct. Here $\sigma$ is the sum-of-divisors function. Distinctness is known for…

Let $A$, $B$, and $C$ be positive integers such that $A+B=C$ and $(A,B)=1$. Define $\operatorname{rad}(ABC)$ to be the product of the distinct prime divisors of $ABC$. Explicit ABC…

Let $n$ be an even perfect number represented as $n=x^m+y^m$, where $x$ and $y$ are positive integers and $m$ is an integer with $m\geq 2$. Makowski's conjecture. Then $m=3$ and…

Classification conjecture for higher cyclotomic fields. The even perfect integers in $\mathbb{Z}[\zeta_p]$ are exactly

Parker, Rushall and Hunt's conjecture. Every odd norm-perfect Eisenstein integer has the form

Unequal-GCD conjecture. Unconditionally, these two quantities are unequal:

Odd-perfect-number nonexistence conjecture. No odd perfect numbers exist.

Let $\mathcal{S}$ be the set of odd positive integers $n$ such that $2n/\sigma(n)-1=1/x$ for some positive integer $x$. For a set of nonnegative integers, its Schnirelmann density…

Let $\mathcal{S}$ be the set of odd positive integers $n$ such that $2n/\sigma(n)-1=1/x$ for some positive integer $x$. Its asymptotic density is the limit, when it exists, of … wh…

Let $\mathcal{S}$ be the set of odd positive integers $n$ such that $2n/\sigma(n)-1=1/x$ for some positive integer $x$, and let $\pi_{\mathcal{S}}(n)$ denote the number of elements…