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Every quasiperfect number has at least eight distinct prime factors

This paper proves that any quasiperfect number must have at least eight distinct prime factors, thereby raising the lower bound from seven by eliminating all potential counterexamples through a combination of elementary algebraic lemmas and rigorously verified computational searches.

Original authors: Akira Toyohara, Ye Tao, Siqiong Yao

Published 2026-08-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Akira Toyohara, Ye Tao, Siqiong Yao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to find a very specific, invisible treasure hidden inside a giant, infinite library. This treasure is a special kind of number called a "quasiperfect number." To understand the hunt, you first need to know what makes a number "perfect." A perfect number is like a perfectly balanced scale: if you add up all the smaller numbers that divide into it (its "divisors"), the total equals the number itself. For example, the number 6 is perfect because its divisors are 1, 2, and 3, and $1 + 2 + 3 = 6$ . It's a mathematical sweet spot.

Now, imagine a number that is just one step off that perfect balance. If you add up all its divisors, the total is exactly one more than twice the number itself. Mathematicians call this a "quasiperfect number." It's the "almost perfect" cousin. The big mystery is: do these numbers even exist? No one has ever found one, and for decades, no one could prove they don't exist either. It's like searching for a ghost in a house that might be empty; you need to be absolutely sure you've checked every single corner before you can say, "No ghosts here." This paper is the story of a massive, high-tech sweep of that house, proving that if such a number exists, it must be incredibly complex, hiding behind a wall of at least eight different prime building blocks.

The Great Hunt for the "Almost Perfect" Number

For a long time, mathematicians knew that if a quasiperfect number exists, it has to be a very strange creature: it must be an odd number and a perfect square. They also knew it had to be built from a certain number of unique prime "ingredients" (like 3, 5, 7, etc.). Before this paper, the best rule they had was that any such number needed at least seven different prime ingredients. This rule had stood since 1982, but it was stuck. The search was like trying to find a needle in a haystack that kept growing. The "haystack" was a list of possible numbers, and for some of the deepest, most complex candidates, the list was so long that computers would have to run for thousands of years just to check them one by one. It was a "non-terminating search"—a loop that never ended.

This paper breaks that loop. The authors, using a mix of clever math tricks and powerful computers, have proven that no quasiperfect number can exist with only seven (or fewer) prime ingredients. If one exists, it must have at least eight distinct prime factors. This is the first time this specific rule has been improved in 44 years.

How They Did It: The Three Magic Keys

The authors didn't just brute-force their way through the problem; that would have taken too long. Instead, they invented three "magic keys" (mathematical lemmas) that turned an impossible, infinite search into a finite, solvable puzzle. Think of it like trying to find a specific person in a stadium of billions.

The Discriminant Key (The "Instant ID"):
Normally, to find the missing piece of a quasiperfect number, you'd have to guess a prime number and then search for a matching partner. It's like guessing a lock combination and trying every key. The first key, based on a high-school algebra formula, changes the game. Instead of searching for the partner, it calculates a specific "fingerprint" (called a discriminant). If the fingerprint isn't a perfect square, you know instantly that the combination is wrong. This turns a massive search into a simple check.
The Sieve Key (The "Security Guard"):
Even with the first key, there are still millions of candidates. The second key acts like a super-efficient security guard at the stadium entrance. It uses "quadratic residues" (a fancy way of checking if a number behaves like a square in a specific math world) to filter out huge groups of candidates at once. If a number fails this test, the guard throws it out immediately without ever checking the rest of its details. This eliminates about 99.999999% of the possibilities before the computer even starts the heavy lifting.
The Resolver Key (The "Time Machine"):
The biggest problem was that some numbers could have exponents (powers) that went on forever. Checking them one by one would take forever. The third key realizes that these infinite powers follow a predictable pattern. Instead of counting up 1, 2, 3... to infinity, this key solves a single equation that tells you exactly which powers are possible. It turns an infinite loop into a single, quick calculation.

The Result: Closing the Case

Using these three keys, the authors ran a massive computation. They had to check 381 different "stems" (starting patterns of prime numbers) which expanded into a staggering 79,751,212 "deep leaves" (the final, complex candidates).

The Outcome: They found zero quasiperfect numbers.
The Proof: The search didn't just stop; it was verified in multiple ways. They ran the calculation on different types of computers (CPUs and GPUs), used different mathematical methods to double-check the results, and even planted fake "solutions" into the code to make sure their system would catch them. Every time, the system correctly said "No solution found."
The Conclusion: They proved that the "deep leaves" of the search tree are all dead ends. The paper explicitly rules out the possibility of a quasiperfect number having 7 or fewer prime factors.

Why This Matters

This isn't just about finding a number; it's about understanding the rules of the universe of numbers. By proving that any quasiperfect number must have at least eight distinct prime factors, the authors have pushed the boundary of what we know. They cleared a massive obstruction that had blocked progress since 1982.

The paper also addresses a previous attempt by another researcher (Zemann) that claimed to find the same result. The authors carefully audited that work and found a small "gap" in the code where 35 possible cases were skipped. Their work is the first to completely close that gap, ensuring the proof is watertight.

In short, the authors have built a fortress of logic and computation. They have shown that if a quasiperfect number is hiding, it is hiding behind a wall of at least eight prime ingredients, making it even more elusive than we thought. The hunt continues, but the map has just been updated with a much larger "Do Not Enter" zone.

Technical Summary: "Every quasiperfect number has at least eight distinct prime factors"

Problem Statement
The paper addresses the existence of quasiperfect numbers, defined as integers $n$ satisfying the equation $\sigma(n) = 2n + 1$ , where $\sigma(n)$ is the sum of all positive divisors of $n$ . While no such number has ever been found, and their non-existence remains unproven, structural constraints are known: any quasiperfect number must be an odd perfect square. A central open question concerns the lower bound on $\omega(n)$ , the number of distinct prime factors of $n$ . Since 1982, the established lower bound has been $\omega(n) \ge 7$ (Hagis–Cohen). This paper aims to advance that bound.

Methodology
The proof proceeds by eliminating all possible cases where $\omega(n) \le 7$ . The strategy involves a computational search tree (the "corridor" method) where the abundancy $\sigma(n)/n$ must lie in the narrow interval $(2, 2 + 1/n]$ . The search space is divided into "stems" (fixed initial primes) and "leaves" (terminal configurations with free primes).

The core innovation is a system of three lemmas at the level of elementary algebra that transform an unbounded, non-terminating search into a finite decision process. These lemmas address three specific dimensions of the search space that previously obstructed termination:

Lemma 1 (Discriminant Criterion): Transforms the search for the last free prime $q$ into a verification problem. By treating the leaf equation as a quadratic in $q$ , the authors derive a discriminant $\Delta(p)$ . A solution exists only if $\Delta(p)$ is a perfect square. This eliminates the need to search a range for $q$ , reducing the task to checking if a specific value is a square.
Lemma 2 (Quadratic-Residue Sieve): Transforms verification into a table lookup. Since $\Delta(p)$ must be a perfect square, it must be a quadratic residue modulo any integer $r$ . By precomputing allowed residue classes for various moduli, the authors can exclude entire classes of intermediate primes $p$ without performing expensive exact arithmetic. This acts as a "coarse sieve" that drastically reduces the candidate set.
Lemma 3 (Multilinear Resolver): Resolves the issue of unbounded exponents. Previously, increasing exponents created an infinite sequence of equations. The authors observe that the sum-of-divisors function for a prime power is linear in the variable $V = p^{2e}$ . This allows the unbounded exponent dimension to be solved directly via linear or Möbius transformations, converting an infinite iteration into a finite set of algebraic solutions.

Two additional "kill lemmas" (Lemma 4 and Lemma 5) are employed for specific deep cases:

Lemma 4 (FULLKILL 9): Uses periodicity modulo 9 to eliminate entire families of exponent configurations if the discriminant is a non-residue for all periods.
Lemma 5 (Tight-Kill): Uses monotonicity arguments to prove that if the minimal configuration of a branch exceeds the corridor upper bound, the entire cone of higher exponents is invalid.

Key Contributions and Results

Main Theorem (Theorem 1): The paper proves that if a quasiperfect number exists, it must satisfy $\omega(n) \ge 8$ . This is the first improvement on the $\omega \ge 7$ bound since 1982.
Elimination of $\omega = 7$ :
- Case $3 \nmid n$ : The search space collapses to a single stem $(5, 7, 11, 13)$ , which is eliminated in under one second.
- Case $3 \mid n$ : This constitutes the bulk of the work, involving 381 stems expanding to 79,751,212 deep leaves. The authors successfully eliminate all these leaves with zero solutions found.
Computational Verification: The elimination is supported by a rigorous four-layer trust framework:
1. Exact Arithmetic: All kills are based on integer arithmetic; no floating-point numbers are used for final decisions.
2. Planted Solutions: The system correctly identifies 12 hand-constructed true solutions in the pilot phase and 24 planted true solutions at the CPU gate plus 24 at the GPU gate (48 total) in the full campaign, while rejecting perturbed false solutions. This ensures the sieve does not kill by mistake and the final check does not let one through.
3. Dual Implementation: Two independent architectures (CPU and GPU) produce identical results.
4. Formal Verification: The lemmas and logic are formalized in the Lean proof assistant (259 theorems, zero sorry), ensuring the logical soundness of the elimination rules.
Audit of Prior Work: The authors audit a 2023 preprint by Zemann claiming the same bound. They identify a "coverage gap" where 35 feasible exponents were skipped due to an off-by-one error in the code, concluding that Zemann's work does not constitute a complete proof, whereas the current paper's elimination is complete.

Significance and Claims
The paper claims to provide the first complete proof that $\omega(n) \ge 8$ for any quasiperfect number. The significance lies not in the discovery of a quasiperfect number, but in the rigorous elimination of the $\omega=7$ case, which had been an obstruction for over 40 years.

The authors emphasize that the breakthrough was achieved not by brute-force computing power, but by mathematical insight that converted an undecidable search (due to unbounded exponents and massive scan ranges) into a finite, verifiable decision process. The work is presented as a "self-contained" proof that does not rely on the unverified parts of previous literature (specifically the unpublished $\omega \ge 9$ part of Hagis–Cohen 1982).

Future Work (as stated in the paper)
The paper reports partial progress toward the next bound, $\omega(n) \ge 9$ . A "sequel campaign" has settled the reduction boundary for $\omega=8$ , showing that any quasiperfect number with $\omega=8$ must be divisible by $3 \cdot 5 \cdot 17 = 255$ (Theorem 2). However, the full elimination of the 516 stems for $\omega=8$ is still in progress, with one "monster stem" $(3, 5, 19, 97)$ remaining as a significant computational hurdle. The authors also identify an open algorithmic problem (P2-EXACT) regarding the counting of lattice points under a polynomial, which they believe is the key to overcoming the remaining computational barriers.

The Great Hunt for the "Almost Perfect" Number

How They Did It: The Three Magic Keys

The Result: Closing the Case

Why This Matters

Technical Summary: "Every quasiperfect number has at least eight distinct prime factors"

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