Erdoes #1052 - working report - Erdős Problem a Day

Erdős problem 1052 — wave w029

Date: 2026-07-29 UTC

0. Mandatory live-page gate

I fetched the live problem page through the Bright Data browser path, not by direct curl.

Live state observed:

OPEN - $10.
0 comments and 0 claimed proofs.
“Interested in collaborating”: none.
“Currently working on this problem”: none.
The page was last edited 28 September 2025.

Thus there was neither a claimed resolution nor a current worker, and the task passed the no-collision gate.

The exact live statement is as follows. The first and last sentences below are verbatim; the middle defining sentence is rendered by the identical formula.

A unitary divisor of $n$ is $d\mid n$ such that $(d,n/d)=1$.
\[ > n\geq 1,\qquad > n=\sum_{\substack{d\mid n\\(d,n/d)=1\\d<n}}d. > \]
Are there only finite many unitary perfect numbers?

The live page additionally reports, as known results, that there are no odd unitary perfect numbers and that the five known examples are

6,\quad 60,\quad 90,\quad 87360,\quad 146361946186458562560000.

It attributes the problem and prize to Guy [Gu04], and links OEIS A002827. I treat those live-page statements, rather than the stale tracker YAML, as the task ground truth.

1. Result obtained

Component-box classification (d: computational-only). Let

n=\prod_{i=1}^k q_i,\qquad q_i=p_i^{e_i},

be the decomposition of $n$ into prime-power components, with the $p_i$ distinct. An exhaustive exact-integer CP-SAT computation gives:

If every component $q_i\leq 10{,}000{,}000$, then $n$ is one of the five known unitary perfect numbers.

Equivalently, any sixth unitary perfect number must have at least one prime-power component

\max_{p^e\parallel n}p^e>10^7.

This is substantially more informative than an integer scan $n\leq X$: the fifth known number is about $1.46\cdot 10^{23}$, yet it lies inside this component box because its largest component is $2^{18}=262144$. The computation permits arbitrarily large products $n$ and does not impose a bound on the number of selected components.

This does not prove the requested finiteness. The classification is a finite-box, solver-dependent certificate and is therefore deliberately labelled (d), not (a).

2. Exact reduction used by the computation

For a prime power $q=p^e$,

\sigma^*(q)=q+1.

Multiplicativity over distinct prime bases gives

\sigma^*(n)=\prod_i(q_i+1).

Consequently,

n\text{ is unitary perfect} \quad\Longleftrightarrow\quad \prod_i(q_i+1)=2\prod_iq_i. \tag{1}

Taking the valuation at every prime $r$, (1) is equivalent to the sparse 0–1 system

\sum_{q\leq B} \left(v_r(q+1)-v_r(q)\right)x_q=v_r(2), \tag{2}

where $q$ ranges over prime powers, $x_q\in\{0,1\}$, and

\sum_{e:\,p^e\leq B}x_{p^e}\leq1 \tag{3}

for every prime base $p$.

Soundness and completeness of (2)–(3) (a: elementary-rigorous).

Any unitary perfect $n$ whose components are at most $B$ sets exactly

its component variables to 1. Equation (1), hence every equation (2), holds, and distinct prime bases give (3).

Conversely, any 0–1 solution of (2)–(3) selects powers of distinct primes.

Equality of every prime valuation makes the two positive integers in (1) equal. Their product $n=\prod_{x_q=1}q$ is therefore unitary perfect.

A prime factor of some $q+1$ can be $B<r\leq B+1$. It is still included

in (2); because no component with base $r$ is available, its equation forces every uncancellable occurrence to zero. Thus there is no hidden smoothness assumption.

This is a bijection, not merely a necessary-condition relaxation.

3. Reproducible computation

The standalone re-verifier is erdos1052_wavew029_reverify.py. Its SHA-256 is

1c60e72010e2c34678f588f7dc6edf9f1d82348ea5ed0359a0fbcd21ba281901

Run it from the repository root with:

python3 runs/erdos1052_wavew029_reverify.py

The central model construction is:

for index, (q, base, exponent) in enumerate(
    zip(q_values, bases, exponents)
):
    valuation_terms[base].append((index, -exponent))
    for prime, valuation in factor_from_spf(q + 1, spf).items():
        valuation_terms[prime].append((index, valuation))

variables = [model.new_bool_var(f"x_{q}") for q in q_values]
for prime, terms in valuation_terms.items():
    rhs = 1 if prime == 2 else 0
    model.add(
        sum(coefficient * variables[i] for i, coefficient in terms) == rhs
    )
for indices in indices_by_base.values():
    model.add(sum(variables[i] for i in indices) <= 1)

The script refuses to report completeness unless OR-Tools returns OPTIMAL after enumerating all solutions. It then performs two arithmetically independent checks:

For every returned component tuple, it generates all ordinary divisors

of $n$, filters them by the original condition $\gcd(d,n/d)=1$, and checks that the proper unitary divisors sum to $n$.

It factors and tests every integer $n\leq100000$ independently, finding

$6,60,90,87360$.

Environment and exact output from the saved script:

Python 3.12.3; OR-Tools 9.15.6755
component bound B=10000000
SPF sieve through 10000001: 3.29s, peak_rss=503 MiB
prime-power variables=665134, valuation primes=664579,
  build-data time=10.95s, peak_rss=860 MiB
CP-SAT model complete: 31.37s, peak_rss=1452 MiB
solver_status=OPTIMAL, solve_time=10.42s, total_time=42.74s,
  branches=4574, conflicts=154, peak_rss=1452 MiB
independent direct scan n<=100000: [6, 60, 90, 87360]
VERIFIED_COMPONENT_BOX: B=10000000;
  integers=[6, 60, 90, 87360, 146361946186458562560000]
ELAPSED=44.94 MAXRSS_KB=1487448

The exact returned table was:

| $n$ | sorted components $q_i$ | $\sigma^*(n)$ | number of unitary divisors | |---:|---|---:|---:| | 6 | $2,3$ | 12 | 4 | | 60 | $3,4,5$ | 120 | 8 | | 90 | $2,5,9$ | 180 | 8 | | 87360 | $3,5,7,13,64$ | 174720 | 32 | | 146361946186458562560000 | $3,7,11,13,19,37,79,109,157,313,625,262144$ | 292723892372917125120000 | 4096 |

The fifth row in particular recomputes

146361946186458562560000 =2^{18}\cdot3\cdot5^4\cdot7\cdot11\cdot13\cdot19 \cdot37\cdot79\cdot109\cdot157\cdot313.

4. A clean global reduction

The following isolates the uniformity gap without computation.

Fixed-component finiteness (a: elementary-rigorous). For every fixed $k$, there are only finitely many unitary perfect numbers with exactly $k$ distinct prime factors.

Proof. Suppose instead that $n_j$ were infinitely many distinct such numbers. Write each $n_j$ as a product of its $k$ prime-power components and order those components increasingly. Passing to diagonal subsequences, each component position is either eventually constant or tends to infinity. Let $C$ be the product of the eventually constant components and $D_j$ the product of the remaining components. Then

\frac{\sigma^*(D_j)}{D_j} =\prod_{q\parallel D_j}\left(1+\frac1q\right)\longrightarrow1.

Since every $C D_j$ is unitary perfect,

2=\frac{\sigma^*(C)}C\frac{\sigma^*(D_j)}{D_j},

so $\sigma^*(C)/C=2$. If $C=1$, this says $1=2$, impossible. If $C>1$, then $C$ itself is unitary perfect. If $D_j=1$ infinitely often, then $n_j=C$ infinitely often, contrary to distinctness. Otherwise pass to a tail on which every $D_j>1$; then

\frac{\sigma^*(CD_j)}{CD_j} =2\frac{\sigma^*(D_j)}{D_j}>2,

again a contradiction. ∎

This re-proves Theorem 4 of Subbarao–Warren (1966).

Exact equivalence (a: elementary-rigorous).

\{\text{unitary perfect numbers}\}\text{ is finite} \quad\Longleftrightarrow\quad \omega(n)\text{ is uniformly bounded on that set}. \tag{4}

The forward implication is immediate. For the reverse implication, take the finite union of the fixed-$k$ sets above.

Thus one exact missing global lemma is a uniform upper bound for the number of components. Existing results give strong lower bounds on the components of a hypothetical sixth example, but no upper bound uniform in all examples. The finite-box computation addresses component size, not this uniformity, so it cannot be promoted to a proof of finiteness.

5. Primary-source literature audit

I searched by exact title, author, DOI, and the phrases “unitary perfect number(s),” “squarefree odd part,” and “largest odd component.” The following are the directly relevant primary sources I could verify.

M. V. Subbarao and L. J. Warren,

“Unitary Perfect Numbers”, Canadian Mathematical Bulletin 9 (1966), 147–153. They prove that no odd unitary perfect number exists, prove finiteness with a fixed $2$-component, and prove finiteness with a fixed number of prime factors. (b: rigorous modulo the cited paper; the last result is also proved from scratch in §4.)

C. R. Wall,

“The Fifth Unitary Perfect Number”, Canadian Mathematical Bulletin 18 (1975), 115–122. It proves that the large displayed example is the next unitary perfect number after 87360. (b)

C. R. Wall,

“On the Largest Odd Component of a Unitary Perfect Number”, Fibonacci Quarterly 25 (1987), 312–316. Apart from the five known examples, the largest odd prime-power component exceeds $2^{15}$. (b)

C. R. Wall,

“New Unitary Perfect Numbers Have at Least Nine Odd Components”, Fibonacci Quarterly 26 (1988), 312–317. Any further example has more than eight odd components. (b)

S. W. Graham,

“Unitary Perfect Numbers with Squarefree Odd Part”, Fibonacci Quarterly 27 (1989), 317–322. If the odd part is squarefree, the only examples are $6,60,87360$. (b)

The 2025 paper by Shen and Zhao, “Unitary Perfect Numbers and Fibonacci Primes”, studies higher-power unitary-divisor equations; its abstract makes no claim to settle the present $a=1$ finiteness question. I found no other verified post-1989 primary paper claiming a resolution.

Audit of arXiv:2605.20475

A directly relevant preprint appeared after the live page's last edit: T. Maciejewski, “Bounded-box reductions in the Subbarao–Warren problem for unitary perfect numbers”, arXiv:2605.20475v2 (24 May 2026). The arXiv record exists, supplies source and ancillary files, and explicitly says that it does not prove finiteness.

I downloaded and inspected its v2 TeX source. Its proposed connection from unitary perfect numbers to “3-Higgs” primes has a fatal gap in the supplied proof. For an odd exponent $e$, that proof treats

q\mid p-1\quad\Longrightarrow\quad q\mid p^e+1.

For every odd $q\mid p-1$, however,

p^e+1\equiv 1^e+1\equiv2\pmod q,

so the asserted divisibility goes in the wrong direction. The smallest explicit audit is

3\mid(7-1),\qquad 3\nmid(7^1+1).

The saved re-verifier recomputes this residue. The proof also invokes exponent caps imposed only inside its bounded kernel box while claiming a statement about arbitrary UPN components. Therefore:

the displayed modular flaw is (a: elementary-rigorous);
the proposition “every UPN prime is 3-Higgs” remains

(c: structural-unverified) on the evidence supplied there;

downstream filters that reject a UPN merely because a seed has a

non-3-Higgs factor do not presently constitute a verified result about all unitary perfect numbers;

the preprint's independent finite computations concerning its auxiliary

set may still be computations about that set, but I do not use them in the component-box result above.

This is why I do not report the preprint as an advance that changes the live problem status.

6. Precise wall and cost of extending this computation

The computation certifies a finite component box only. Repeating it for larger and larger finite $B$ does not supply the uniform finite $B$ needed for the full problem.

Observed scaling on this VM was:

| $B$ | prime-power variables | total time | peak RSS | |---:|---:|---:|---:| | $5\cdot10^6$ | 348940 | 22.45 s | 1126 MiB | | $10^7$ | 665134 | 44.94 s | 1453 MiB |

A roughly linear extrapolation to $B=10^8$ gives about 5.8 million variables, 6–10 minutes, and roughly 13–20 GiB. This is only an engineering estimate (c); solver search need not scale linearly. I did not run that larger job because it exceeds the requested few CPU-minutes and, more importantly, no finite extension closes the uniformity gap.

The exact missing global step is one of the following equivalent kinds of information:

a uniform upper bound on $\omega(n)$, which closes the problem by (4);
a uniform upper bound on the largest component, which reduces the problem

to one finite system such as (2)–(3);

another theorem that directly prevents an infinite sequence whose number

and size of components both escape.

None of the checked primary sources supplies such a uniform upper bound.

7. Claim ledger

| Claim | Label | Verification | |---|---|---| | Product/valuation reduction (1)–(3) | (a) elementary-rigorous | algebra above; reconstructed exactly in code | | Fixed-$k$ finiteness and equivalence (4) | (a) elementary-rigorous | from-scratch subsequence proof | | Classical structural theorems | (b) rigorous-modulo-named-theorem | linked primary papers | | No extra solution with every component $\le10^7$ | (d) computational-only | CP-SAT OPTIMAL, full enumeration | | All five output rows really satisfy the original gcd definition | (d), exact arithmetic | all ordinary divisors independently generated | | Global finiteness | not proved | exact uniformity gap identified | | 3-Higgs bridge in arXiv:2605.20475v2 | (c) structural-unverified | supplied proof has explicit modular failure |

PARTIAL: Exact exhaustive computation finds only the five known unitary perfect numbers when every prime-power component is at most $10^7$; any sixth has a component $>10^7$, while full finiteness still requires a uniform component-count or component-size bound.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger