On odd deficient-perfect numbers with four distinct prime divisors

Cui-Fang Sun; Zhao-Cheng He

REVIEW 2 major objections 6 minor 11 references

On odd deficient-perfect numbers with four distinct prime divisors

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The only odd deficient-perfect number with four distinct prime divisors is $3^2\cdot 7^2\cdot 11^2\cdot 13^2$, with deficient divisor $3^2\cdot 7\cdot 13$.

desk verdict A workmanlike completion of a four-prime deficiency classification; the internal inequality chase is mostly credible, but the theorem leans on an unreproduced reduction from [6] that a referee must check. read the letter →

arxiv 1908.04932 v1 pith:5KSNP5UT submitted 2019-08-14 math.NT

Cui-Fang Sun , Zhao-Cheng He This is my paper · ORCID

classification math.NT MSC 11A25

keywords deficient-perfectnumbersdeficientdivisordivisor-sumequationoddfourdistinctprimedivisorsmultiplicativearithmeticfunctionclassificationtheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a classification: among odd integers with exactly four distinct prime divisors, the equation $\sigma(n)=2n-d$ has exactly one solution with $d$ a proper divisor of $n$, namely $n=3^2\cdot 7^2\cdot 11^2\cdot 13^2$ with $d=3^2\cdot 7\cdot 13$. Deficient-perfect numbers generalize almost perfect numbers, where the deficit $d$ would have to be $1$; here the deficit may be any proper divisor. Because earlier results ruled out odd deficient-perfect numbers with fewer than four distinct prime factors, the theorem completes the classification up to that point: any odd deficient-perfect number beyond the displayed one must have at least five distinct prime factors. The proof matters because it reduces an infinite-looking divisor-sum condition to a finite case analysis and identifies the single configuration that survives.

What carries the argument

The load-bearing identity is the normalized divisor-sum equation $\sigma(n)/n+d/n=2$, equivalently $\sigma(n)/n+1/D=2$ where $D=n/d$ is the part of $n$ not covered by the deficient divisor. Since $\sigma$ is multiplicative, the left side expands to $\prod_{i=1}^4 \frac{p_i^{\alpha_i+1}-1}{(p_i-1)p_i^{\alpha_i}}+\frac{1}{D}$. The proof treats this identity as a measuring rod: elementary bounds on each factor $\frac{p^{\alpha+1}-1}{(p-1)p^\alpha}$ either push the product above $2$ or show that $1/D$ is too small to close the gap, and when a gap survives, order arguments (the smallest exponent $h$ with $a^h\equiv 1\pmod m$) and quadratic-residue symbols (records of whether a number is a square modulo a prime) produce contradictions. The last remaining configuration is then solved directly.

What would settle it

Check every odd $n$ with exactly four distinct prime divisors by computing $\sigma(n)-2n$: if this value is $-d$ for some proper divisor $d$ of $n$ and $n\neq 3^2\cdot 7^2\cdot 11^2\cdot 13^2$, the theorem is false. A concrete search would enumerate candidates of the form $3^a p^b q^c r^d$ with $a,b,c,d$ even and small, and primes $p,q,r$ in the ranges the proof eliminates, checking whether $\sigma(n)=2n-d$ holds directly.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: the only odd deficient-perfect number with four distinct prime divisors is $3^2\cdot 7^2\cdot 11^2\cdot 13^2$, with deficient divisor $3^2\cdot 7\cdot 13$. Writing a candidate as $n=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}p_4^{\alpha_4}$, the proof uses cited facts to restrict the shape: the deficient divisor $d$ is larger than $1$, all four exponents $\alpha_i$ are even, and a prior reduction leaves only $p_1=3$ with $p_2\in\{5,7,11,13,17\}$. The remaining cases are eliminated by combining upper and lower estimates for $\sigma(n)/n+d/n=2$ with divisibility conditions forced by multiplicative orders and by quadratic-residue symbols; in the one surviving case the estimates collapse to $\alpha_i=2$ and primes $3,7,11,13$, producing the displayed number and its deficient divisor.

Load-bearing premise

The proof inherits, without re-deriving, cited results that the deficient divisor is nontrivial, that all prime exponents are even, and that an odd deficient-perfect number with four prime divisors must have smallest prime $3$ and second prime in $\{5,7,11,13,17\}$; if any of those cited classifications has an omitted case, the uniqueness conclusion could miss a solution.

Editorial extensions

If this is right

For odd integers with exactly four distinct prime divisors, $\sigma(n)=2n-d$ has exactly one solution: $n=3^2\cdot 7^2\cdot 11^2\cdot 13^2$ with $d=3^2\cdot 7\cdot 13$.
Combined with cited results for one, two, and three prime factors, the theorem implies that every odd deficient-perfect number other than the displayed one has at least five distinct prime divisors.
The case analysis is finite: after the cited reduction, the proof checks the second prime values $5,7,11,13,17$ and eliminates all prime and exponent patterns except $(3,7,11,13)$ with exponents $2$.
The displayed deficient divisor is larger than $1$, matching the cited fact that $d>1$ for odd deficient-perfect numbers, so the unique four-prime example is not an almost perfect number.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not claimed by the paper: the same normalized-equation and bounding strategy could be pushed to $\omega(n)=5$, where it should still shrink the search to finitely many exponent patterns before order and quadratic-residue arguments are applied.
Not claimed by the paper: because the unique solution has all exponents equal to $2$, one could probe whether further odd deficient-perfect numbers, if they exist, have unusually small exponents or a regular shape; the theorem gives no evidence in either direction.
Not claimed by the paper: a direct re-verification of the cited reduction (the list of possible second primes) would make the uniqueness conclusion self-contained and would be the most useful follow-up check of the argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Zhiwei, quick take on Sun–He, arXiv:1908.04932. The title is the theorem: the only odd deficient-perfect number with four distinct prime divisors is 3^2·7^2·11^2·13^2, with deficient divisor 3^2·7·13. That is a real classification result, and I do not see circularity or data-fitting anywhere. Identity (1.1) is correct, the surviving example checks out, and the lemmas are mostly careful bounds showing σ(n)/n dips above or below 2 after pinning down exponents.

What is new: the final uniqueness for the four-prime case. The method is not new—it extends the bracket of reductions and inequalities from [5], [6], and [9]—but the endpoint is new and the work is honest. The citation pattern is fine; reliance on prior classifications is normal, not a flaw by itself.

Where the real risk sits is external. The proof of Theorem 1 opens by importing, without restating hypotheses, the reduction from [6] that p1 = 3 and p2 ∈ {5,7,11,13,17}. Every later case assumes that. If [6] has an overlooked case, or was proved under a hypothesis not met here, the uniqueness conclusion misses a solution. The paper also takes d > 1 from [5] and even exponents from [9] without re-derivation. A referee needs to check those three sources against the setting of this paper.

Internal soft spots are minor but real. Lemma 3.2, Case 2 jumps to α1 = 2 after ruling out α1 ≥ 8, without explicitly dispatching α1 = 4 or 6; a short lower-bound inequality fills the gap. In the proof of Theorem 1, Case 4, there is a typo where the text reads “α2 ≥ 12, α2 ≥ 4” and one of those should be α1. There are also some garbled inequality lines in Section 2 where a threshold like p4 ≥ 607 appears without the denominator that follows; these are readable as minor exposition slips, not load-bearing failures.

Net: if the imported [6] reduction is sound, this paper is probably right. It is exactly the kind of specialized, case-heavy result that needs a referee with access to [5], [6], and [9] and the patience to verify a long chain. I would not desk-reject it. Send it out; after the external reduction is confirmed and the small gaps are patched, I would accept.

Referee Report

2 major / 6 minor

Summary. The paper claims to prove Theorem 1.1: the only odd deficient-perfect number with four distinct prime divisors is 3^2·7^2·11^2·13^2, with deficient divisor 3^2·7·13. The proof writes n=∏ p_i^{α_i}, d=∏ p_i^{β_i}, and D=∏ p_i^{α_i−β_i}, and uses the identity σ(n)=d(2D−1) to reformulate the defining equation. It then invokes a reduction from [6] to p1=3 and p2∈{5,7,11,13,17}, and eliminates the remaining cases by a long series of elementary lemmas based on upper and lower bounds for σ(n)/n, multiplicative orders, and Legendre-symbol computations.

Significance. If the theorem is correct, it completes the classification of odd deficient-perfect numbers with up to four distinct prime factors, continuing the line of work in [9], [10], and [11]. The central identity (1.1) is sound, the argument is elementary and parameter-free, and the case analysis is substantial. The main caveat is that the proof is conditional on an unstated reduction from [6]; the value of the paper depends on that reduction being available and correctly quoted. No machine-checked verification is supplied, but the arithmetic claims are explicit and checkable.

major comments (2)

[§5, Proof of Theorem 1] The proof begins with 'By [6], we need to consider p1=3 and p2∈{5,7,11,13,17}', and every subsequent case split depends on this reduction. The manuscript neither states the theorem of [6] nor verifies that the hypotheses under which [6] applies (oddness, four distinct prime factors, the definition of deficient divisor, or possible restrictions on D) match the present setting. If [6] was proved under a different convention, the uniqueness conclusion could miss a valid solution. Please state the quoted result precisely and either supply a proof or give a detailed reference with the exact statement; an appendix re-deriving this reduction would make the paper self-contained.
[§3, Lemma 3.2, Case 2] In Case 2 (D=9), the proof rules out α1≥8 by a lower bound and then concludes 'Thus p4≥103 and α1=2'. The displayed reasoning does not exclude α1=4 or α1=6, and the preceding sentence 'Since ord17(7)=16, ord17(13)=4, we have p4≡1 (mod 17)' does not follow from those stated orders as written. A missing congruence or divisibility argument is needed here; as it stands, this subcase is incomplete and Lemma 3.2 is not fully proved.

minor comments (6)

[§1, Introduction] The statement 'By [5] and [9], we have d>1 and α_i's are all even' should attribute evenness to the elementary parity argument for odd n and reserve [5] for the d>1 assertion.
[§2, Lemma 2.7, Case 5] The notation 'D = {41,47}' should be 'D ∈ {41,47}'.
[§5, Case 4] After the list p4∈{1021,1531,2551,3061}, the text reads 'It follows that α2≥12, α2≥4 and α3≥4'; one of the first two inequalities is a typo, presumably α1≥12.
[§3, Lemma 3.6] In equation (3.6), the first factor is written as '3^{α2+1}−1'; it should be '3^{α1+1}−1'.
[General exposition] Many congruence deductions are highly compressed, e.g., 'Since ord_m(a)=r, we have ...' without displaying the resulting factorization; adding the explicit divisibility statements would make the case analysis much easier to verify.
[References] Reference [1] appears unrelated to deficient-perfect numbers; if it is not actually used, it should be removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is a direct case analysis from the defining equation, with external cited reductions as normal support.

full rationale

The paper derives Theorem 1.1 from the definition sigma(n)=2n-d, rewritten as equation (1.1), and then proceeds by explicit case analysis over the possible primes and exponent ranges. No parameter is fitted to data, no quantity is defined in terms of the quantity it is supposed to predict, and the unique solution 3^2*7^2*11^2*13^2 emerges from the inequalities and order arguments in Lemma 3.1 rather than from an ansatz. The cited premises - d>1 and even exponents from [5] and [9], and the reduction to p1=3 with p2 in {5,7,11,13,17} from [6] - are prior results by other authors, not self-citations, and they are used as lemmas; even the even-exponent fact is independently immediate from parity for odd n. Whether the classification in [6] is complete is an external correctness question, not a circularity question. Because none of the paper's claims reduces by construction to its own inputs, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard multiplicative properties of sigma, on two quoted reductions from the literature ([5] and [9] for d greater than 1 and even exponents, [6] for p1 = 3 and p2 in {5, 7, 11, 13, 17}), and on the verification that the listed number satisfies sigma(n) = 2n minus d. There are no free parameters fitted to data and no invented entities.

assumptions (3)

standard math Multiplicativity of sigma and the sum-of-divisors formula sigma(p^alpha) = (p^(alpha+1) - 1)/(p - 1) are standard and used throughout (Eq. 1.1).
The identity (1.1) is the foundation of every lemma; it follows from multiplicativity, not from fitting to data.
domain assumption Prior results [5] and [9]: for odd deficient-perfect n with four distinct prime factors, d is greater than 1 and all exponents alpha_i are even.
Stated in the introduction as inherited from the literature; if false, the parity and d greater than 1 restrictions used throughout the case analysis collapse.
domain assumption Prior result [6]: p1 = 3 and p2 is in {5, 7, 11, 13, 17}.
Used at the start of Proof of Theorem 1 to reduce to the cases examined in Sections 2 through 4; the current paper does not re-derive this reduction.

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Pith. "Pith review of On odd deficient-perfect numbers with four distinct prime divisors." pith.science (2026). https://pith.science/paper/5KSNP5UT

@misc{pith2026190804932,
  author       = {Pith},
  title        = {Pith review of: On odd deficient-perfect numbers with four distinct prime divisors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KSNP5UT}},
  note         = {Machine review of arXiv:1908.04932}
}

abstract

For a positive integer $n$, let $\sigma(n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $\sigma(n)=2n-d$. In this paper, we show that the only odd deficient-perfect number with four distinct prime divisors is $3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}$.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[6]

X. Y . Ma and Y . J. Wang, On deﬁcient-perfect numbers with four distinct prime divis ors, Pure Appl. Math. 31(2015), 643-649

work page 2015
[5]

Kishore, Odd integers n with ﬁve distinct prime factors for which 2 − 10− 12 < σ (n)/n < 2 + 10 − 12, Math

M. Kishore, Odd integers n with ﬁve distinct prime factors for which 2 − 10− 12 < σ (n)/n < 2 + 10 − 12, Math. Comp. 32 (1978), 303309

work page 1978
[9]

Tang and M

M. Tang and M. Feng, On deﬁcient-perfect numbers , Bull. Aust. Math. Soc. 90(2014), 186194

work page 2014
[10]

M. Tang, X. Y . Ma and M. Feng, On near-perfect numbers, Colloq. Math. 144(2016), 157-188

work page 2016
[11]

M. Tang, X. Z. Ren and M. Li, On near-perfect and deﬁcient-perfect numbers , Colloq. Math. 133(2013), 221- 226

work page 2013
[1]

Ahn, Y .S

J. Ahn, Y .S. Shin, The minimal free resolution of a star-c onﬁguration in Pn and the weak-Lefschetz property. J. Korean Math. Soc. 49 (2012), 405–417

work page 2012
[2]

Anavi, P

A. Anavi, P . Pollack and C. Pomerance, On congruences of the form σ (n) ≡ a (mod n), Int. J. Number Theory 9 (2013), 115124

work page 2013
[3]

G. L. Cohen, On odd perfect numbers (II), multiperfect numbers and quasi perfect numbers, J. Aust. Math. Soc. 29 (1980), 369384

work page 1980

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P . Hagis and G. L. Cohen, Some results concerning quasiperfect numbers , J. Aust. Math. Soc. 33(1982), 275286

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

X. Y . Ma and Y . J. Wang, On deﬁcient-perfect numbers with four distinct prime divis ors, Pure Appl. Math. 31(2015), 643-649

work page 2015

[2] [5]

Kishore, Odd integers n with ﬁve distinct prime factors for which 2 − 10− 12 < σ (n)/n < 2 + 10 − 12, Math

M. Kishore, Odd integers n with ﬁve distinct prime factors for which 2 − 10− 12 < σ (n)/n < 2 + 10 − 12, Math. Comp. 32 (1978), 303309

work page 1978

[3] [9]

Tang and M

M. Tang and M. Feng, On deﬁcient-perfect numbers , Bull. Aust. Math. Soc. 90(2014), 186194

work page 2014

[4] [10]

M. Tang, X. Y . Ma and M. Feng, On near-perfect numbers, Colloq. Math. 144(2016), 157-188

work page 2016

[5] [11]

M. Tang, X. Z. Ren and M. Li, On near-perfect and deﬁcient-perfect numbers , Colloq. Math. 133(2013), 221- 226

work page 2013

[6] [1]

Ahn, Y .S

J. Ahn, Y .S. Shin, The minimal free resolution of a star-c onﬁguration in Pn and the weak-Lefschetz property. J. Korean Math. Soc. 49 (2012), 405–417

work page 2012

[7] [2]

Anavi, P

A. Anavi, P . Pollack and C. Pomerance, On congruences of the form σ (n) ≡ a (mod n), Int. J. Number Theory 9 (2013), 115124

work page 2013

[8] [3]

G. L. Cohen, On odd perfect numbers (II), multiperfect numbers and quasi perfect numbers, J. Aust. Math. Soc. 29 (1980), 369384

work page 1980