33 problems

Generalized perfect-number conjecture. One has

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…

Odd perfect number conjecture. There are 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:

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

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…

Generalized Sylvester conjecture. There is no odd $m$-perfect number for $m\geq 2$.

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

Primitive non-deficient number component conjecture. There is an $i$, with $1\leq i\leq k$, such that

Let $p$ and $q$ be a $\sigma_{2,2}$ pair, meaning that they are distinct odd primes satisfying the paper's definition of a $\sigma_{2,2}$ pair. Squarefreeness conjecture. The integ…

For a positive integer $k\geq 3$, consider odd deficient perfect numbers whose prime factorization has exactly $k$ distinct prime factors. Finiteness conjecture. For any positive i…

An odd deficient perfect number is an odd positive integer whose proper-divisor sum is less than the number itself; it has four distinct prime factors when its prime factorization…

Odd norm-perfect form conjecture. The integer $\alpha$ must have the form

Let $n=km>1$ be a spoof odd perfect number. If $\gcd(m,k)=1$, then $\sqrt{k}$ is an integer and the sum-of-divisors function $\sigma$ is evaluated at this factor. Square-root abund…

Let $n=km>1$ be a spoof odd perfect number, where $I(x)=\sigma(x)/x$ is the abundancy index. Abundance and factor-ordering conjecture. The following conditions hold: - $I(n)>2$, so…