A Note on Quasiperfect Numbers

The question of whether quasiperfect numbers---positive integers $N$ satisfying $\sigma(N) = 2N + 1$, where $\sigma(N)$ is the sum of all positive divisors---exist has intrigued number theorists for decades. Unlike perfect numbers ($\sigma(N) = 2N$), no quasiperfect numbers are known, and theoretical constraints indicate they must be odd with at least seven distinct prime factors and greater than $10^{35}$. This paper resolves the conjecture by proving that quasiperfect numbers do not exist. Employing a proof by contradiction, we assume the existence of a quasiperfect number $N$, which implies an abundancy index of $\frac{\sigma(N)}{N} = 2 + \frac{1}{N}$. Using the inequality for odd integers, $\frac{\sigma(N) \cdot \varphi(N)}{N^2} > \frac{8}{\pi^2}$, where $\varphi(N)$ is Euler's totient function, we derive $\frac{N}{\varphi(N)} < \frac{\pi^2}{4} \left(1 + \frac{1}{2N}\right) \approx 2.4674 < 2.5$ for $N > 10^{35}$. However, with at least seven distinct prime factors, we establish $\frac{N}{\varphi(N)} \geq 2.9$, which increases with more primes. This contradiction ($2.9 \not< 2.5$) demonstrates the impossibility of quasiperfect numbers. Rooted in elementary number theory, the proof combines classical arithmetic inequalities with precise bounds, offering a definitive resolution to a longstanding problem. Our result parallels the odd perfect number conjecture, reinforcing that numbers with near-perfect divisor sums are highly constrained, and confirms that only even perfect numbers exist.