On odd perfect numbers with exactly one even exponent greater than 2

Technical Summary: On Odd Perfect Numbers with Exactly One Even Exponent Greater Than 2

Problem Statement
This paper investigates the existence and properties of "restrained" odd perfect numbers (OPNs). An OPN is an odd integer $N$ satisfying $\sigma(N) = 2N$, where $\sigma$ is the sum-of-divisors function. Euler established that any OPN must take the form $N = p^e m^2$, where $p$ is a prime (the special prime) with $p \equiv e \equiv 1 \pmod 4$, and $p \nmid m$.

McDaniel previously demonstrated that $m$ cannot be square-free, implying that at least one even exponent in the prime factorization of $N$ must be at least 4. The authors define an OPN as restrained if it possesses exactly one such exponent greater than 2. Let $n^q$ denote the unique component where $n \neq p$ and $q \ge 4$; $n$ is termed the "notable prime" and $q$ the "notable exponent." The primary objective is to establish a lower bound for $q$ under the assumption that a restrained OPN exists.

Methodology
The authors employ a combination of number-theoretic lemmas, computational sieving, and abundancy analysis to rule out specific configurations of prime factors.

1. Structural Lemmas:

   • Modular Constraints: The paper utilizes properties of the multiplicative order ($\text{ord}_s(n)$) and quadratic residues to constrain the values of primes. For instance, Lemma 6 proves that the special prime $p$ must satisfy $p \equiv 1 \pmod 3$.
   • Divisibility Rules: Lemma 4 and Lemma 5 establish conditions under which a prime $s$ divides $\sigma(n^q)$ or $\sigma(t^v)$ based on the parity of orders and the relationship between distinct primes.
   • Exclusion of Solutions: Lemma 3 (derived from Bennett and Levin) asserts that $z^2 + z + 1 = r^t$ has no solution for prime $z, r$ and $t > 1$. This is used to limit the number of primes $s$ such that $\sigma(s^2)$ is a power of a single prime.

2. Case Analysis on the Prime 3:
   The proof of the main theorem is divided into three exhaustive cases regarding the exponent of the prime 3 in $N$:

   • Case 1: 3 is not a factor. The authors show that if $3 \nmid N$, the abundancy index $\sigma(N)/N$ cannot reach 2. This is achieved by bounding the contribution of "ordinary" components (primes with exponent 2) and "sporadic" components using a factor tree approach and computational verification of small primes.
   • Case 2: 3 is ordinary ($3^2 \parallel N$). Using a factor tree derived from the component $3^2$, the authors systematically rule out potential notable primes ($n$) from a set $S = \{7, 13, 19, 31, 61, 97, 127, 331\}$. For each candidate $n$, they demonstrate that the resulting configuration of required prime factors leads to an abundancy strictly less than 2 or creates a contradiction (e.g., requiring more than two copies of 3).
   • Case 3: 3 is notable ($3^q \parallel N$). This is the only remaining viable case. Here, the focus shifts to deriving a lower bound for $q$.

3. Computational Sieving:
   To handle the case where 3 is notable, the authors utilize a C++/GMP program (loopless.cpp) to sieve ordinary primes. The algorithm maintains sets of primes that cannot be ordinary ($M_1$) and primes congruent to $1 \pmod 3$ that cannot divide $N$ ($M_2$). The program iterates through primes to calculate the maximum possible abundancy contribution from ordinary components.

Key Results

• Theorem 1: If $N$ is a restrained odd perfect number, then the notable prime $n$ must be 3, and the notable exponent $q$ satisfies $q \ge 23,000,000,000$.
• Improvement on Bounds: This result significantly improves upon the previously known bound of $q \ge 16$, which was derived from a combination of earlier modular constraints ($q \not\equiv 2 \pmod 3$, $q \neq 4, 6, 10, 12$).
• Reduction to $n=3$: The paper proves that if a restrained OPN exists, the notable prime must be 3. All other candidates for the notable prime are eliminated via abundancy contradictions.
• Lower Bound Derivation: By showing that the abundancy of the special component and the notable component ($3^q$) combined with the maximum possible contribution from ordinary primes (even with over 46 billion ordinary primes) fails to reach 2 unless $q$ is extremely large, the authors derive the bound $q \ge 23,000,000,000$.

Significance and Claims
The paper claims to reduce the existence problem of restrained odd perfect numbers to a single specific case: the notable prime is 3. While the authors do not claim to have disproven the existence of OPNs entirely, they demonstrate that the "restrained" scenario is highly constrained.

The authors modestly note that their bound on $q$ can likely be improved by orders of magnitude using more advanced sieve theory arguments. However, they emphasize that completely ruling out the case where 3 is notable (and thus proving that every OPN must have at least two even exponents greater than 2) requires "new ideas."

The paper concludes by posing two open questions related to the structure of $\sigma(x^2)$ for prime $x$:

1. Whether there exists a prime $x$ such that $x^2 + x + 1$ is square-full.
2. Whether there exist primes $x, y, p, r$ and integers $a > b > 0$ such that $x^2 + x + 1 = p^a r$ and $y^2 + y + 1 = p^b r$.

A negative answer to these questions would allow for tighter constraints on the possible pairs of exponents in the factorization of OPNs, potentially leading to stronger non-existence results. The authors provide their programs and data to facilitate further research in this direction.