Erdős problem #830 — wave 7o
Accessed and checked on 2026-07-27 (UTC). All finite arithmetic in this
report is recomputed by the standalone standard-library checker
runs/erdos830_wave7o_reverify.py.
0. Mandatory live-page audit
I fetched the Cloudflare-protected live page through the Bright Data browser.
The live state was:
- status: OPEN;
- comments: 0;
- claimed proofs: 0;
- “Currently working on this problem”: None;
- “Interested in collaborating”: None;
- all other reaction/worker markers: None;
- last page edit: 28 September 2025.
Thus none of the required stop conditions applies.
The following is the verbatim problem statement from the current page:
> We say that \(a,b\in\mathbb{N}\) are an amicable pair if
> \(\sigma(a)=\sigma(b)=a+b\). Are there infinitely many amicable pairs? If
> \(A(x)\) counts the number of amicable \(1\leq a\leq b\leq x\) then is it
> true that
> \[ > A(x)>x^{1-o(1)}? > \]
The page's verbatim known-results paragraph is:
> For example \(220\) and \(284\). Erdős [Er55b] proved that
> \(A(x)=o(x)\), and Pomerance [Po81] improved this to
> \[ > A(x)\leq x\exp(-(\log x)^{1/3}) > \]
> and later [Po15] to
> \[ > A(x)\leq x\exp\!\left( > -(\tfrac12+o(1))(\log x\log\log x)^{1/2}\right). > \]
> This is problem B4 in Guy's collection [Gu04].
Authoritative page: <https://www.erdosproblems.com/830>. Its LaTeX rendering
is at <https://www.erdosproblems.com/latex/830>.
1. Results obtained
1. Exact finite census — (d) computational-only. For the equation and
weak ordering printed on the live page,
\[ \boxed{A_{\rm literal}(10^8)=236.} \]
Five are diagonal solutions
\[ (6,6),(28,28),(496,496),(8128,8128), (33550336,33550336), \]
and 231 are distinct amicable pairs. If the conventional requirement
\(a\ne b\) is intended, the exact answer is instead
\[ \boxed{A_{\rm distinct}(10^8)=231.} \]
2. Exact normalization — (a) elementary-rigorous. Put
\(s(n)=\sigma(n)-n\). Then the displayed equations are exactly
\[ s(a)=b,\qquad s(b)=a. \tag{1.1} \]
Hence each \(a\) has at most one possible mate, namely \(s(a)\), and
exhaustive enumeration needs no pairwise search. A diagonal solution is
exactly a perfect number. For distinct \(a is deficient. 3. Exact counting reduction — (a) elementary-rigorous. If \[
C(x)=\#\{2\leq n\leq x:s(n)\leq x,\ s(s(n))=n,\ s(n)\ne n\}
\] and \(P(x)=\#\{n\leq x:s(n)=n\}\), then \[
A_{\rm distinct}(x)=\frac{C(x)}2,\qquad
A_{\rm literal}(x)=P(x)+\frac{C(x)}2. \tag{1.2}
\] Every off-diagonal two-cycle contributes its two endpoints to \(C(x)\); every fixed point contributes once to \(P(x)\). 4. **A precise wall for the standard constructive route — (a) elementary-rigorous for the reduction, (c) for prospects beyond it.** The classical Thābit construction reduces infinitude to a simultaneous prime problem stated in Section 6. Even if every admissible exponent worked, that one-parameter family supplies only \(O(\log x)\) pairs up to \(x\), so it cannot establish the conjectured \(x^{1-o(1)}\) lower bound. None of the audited sources supplies the missing simultaneous-primality theorem. The problem remains open. The output here is an exact finite result, a definition-sensitive reduction, and an explicit diagnosis of what the classical machinery fails to provide; it is not an asymptotic lower bound. The live text does not say \(a\ne b\), and it uses \(a\leq b\), so its displayed equation literally admits \(a=b\) whenever \(\sigma(a)=2a\). In contrast: positive integers”; that this excludes perfect numbers. Therefore I report both interpretations rather than silently importing the conventional distinctness condition. This distinction matters for exact counts, although it does not resolve either infinitude question. On amicable numbers, Publ. Math. Debrecen 4 (1955), 108–111. Its theorem says that the set of amicable numbers has density zero. It also explicitly states the infinitude and \(n^{1-\varepsilon}\)-type lower-bound conjecture. Primary scan: <https://users.renyi.hu/~p_erdos/1955-03.pdf>. J. Reine Angew. Math. 325 (1981), 183–188, DOI <https://doi.org/10.1515/crll.1981.325.183>. The cited \(x\exp(-(\log x)^{1/3})\) upper bound is genuine. Author-hosted scan: <https://math.dartmouth.edu/~carlp/Amicable2.pdf>. On amicable numbers, in Analytic Number Theory (2015), 321–327. Author manuscript: <https://math.dartmouth.edu/~carlp/amicablesv3.pdf>. It says that infinitude is unproved and proves a sharper upper bound. H. J. J. te Riele, *Computation of All the Amicable Pairs Below \(10^{10}\)*, Math. Comp. 47 (1986), 361–368 plus supplement, reports an exhaustive algorithm, 236 distinct pairs with smaller member below \(10^8\), and 1427 with smaller member below \(10^{10}\). Primary scan: <https://sech.me/ap/S0025-5718-1986-0842142-3.pdf>. The reciprocal sum of the amicable numbers, Math. Comp. 88 (2019), 1503–1526, explicitly says infinitude is still unknown: <https://math.dartmouth.edu/~carlp/mcom3362.pdf>. Chen--Tang--Zhan, Formalization of Amicable Numbers Theory, arXiv:2601.07444v1 (12 January 2026), formalizes classical construction rules and again lists infinitude as open: <https://arxiv.org/abs/2601.07444>. Exact-title, author-page, arXiv, and recent-year searches found no primary source proving infinitude or the proposed lower bound. This is an honest search result, not a proof that no uncatalogued source exists. The source audit found a real mismatch. The live page attributes to [Po15] But Theorem 1.1 in the cited Pomerance manuscript states, for the number \(\#\mathcal A(x)\) of amicable integers up to \(x\), Thus the cited paper has \(\log\log\log x\), not \(\log\log x\). The harmless pair-count versus member-count convention changes at most a factor of two; it does not explain the missing logarithm. I transcribed the live page exactly in Section 0, but I do not treat its stronger displayed [Po15] formula as source-verified. A broad web search surfaced Surajit Ghosh's Numbers Are 3 Dimensional, <https://rxiv.org/pdf/2004.0294v4.pdf>, whose Theorem 24 asserts infinitude. The displayed argument merely states the conditional Euler/Thābit prime formula and then says a “harmonicity form” makes infinitely many outputs. It never proves that the three required numbers are simultaneously prime for infinitely many parameters. This is exactly the missing lemma in Section 6, so the document does not provide a valid proof and does not change the live OPEN status. Let \(N\) be the requested bound. The main engine sieves the smallest prime factor \(p\) of every \(2\leq n\leq N\). Write \(m=n/p\). \[
\sigma(n)=(p+1)\sigma(m).
\] is \(p^e\), and \[
\sigma(n)=\sigma(m)+p^{e+1}\sigma(r). \tag{4.1}
\] Induction on \(n\) proves that the resulting table contains the exact \(\sigma(n)\) for every \(n\leq N\). The program then checks the unique candidate \(b=\sigma(a)-a\) for every \(a\leq N\), retaining it exactly when \(a\leq b\leq N\) and \(\sigma(b)-b=a\). By (1.1), this loses and adds nothing. These algorithmic completeness statements are (a) elementary-rigorous. The arrays use checked 32-bit unsigned storage. Every computed divisor sum is formed first as a Python integer and triggers an exception rather than wrapping if it exceeds \(2^{32}-1\). The second engine initializes \(s(n)=1\) for \(n\geq2\), \(s(1)=0\), and, for each \(d=2,\ldots,\lfloor N/2\rfloor\), adds \(d\) to Every proper divisor \(d>1\) of every \(n\leq N\) is added exactly once, while the divisor \(1\) was initialized exactly once. This independently constructs the proper-divisor-sum table without the recurrence (4.1). At \(N=10^7\), the two engines produced identical complete ordered pair lists. Every reported endpoint was then recomputed a third way, by trial-division factorization and the prime-power product formula for \(\sigma\). At \(N=10^8\), the faster factorization engine was used and every endpoint was again trial-factor checked. The complete off-diagonal stream also agrees with the published 236-smaller-member census after removing the five pairs whose larger member exceeds \(10^8\). For the stream-level comparison I parsed Hisanori Mishima's public transcription <https://www.asahi-net.or.jp/~kc2h-msm/mathland/math09/math09t1.htm>; te Riele's primary paper supplies the underlying exhaustiveness result, while the transcription was used only as an external sanity check. All rows below are (d) computational-only, with the elementary completeness proof in Section 4. | \(x\) | equation-only \(A_{\rm literal}(x)\) | diagonal/perfect | distinct pairs with both members \(\leq x\) | |---:|---:|---:|---:| | \(10\) | 1 | 1 | 0 | | \(10^2\) | 2 | 2 | 0 | | \(10^3\) | 4 | 3 | 1 | | \(10^4\) | 9 | 4 | 5 | | \(10^5\) | 17 | 4 | 13 | | \(10^6\) | 44 | 4 | 40 | | \(10^7\) | 104 | 4 | 100 | | \(10^8\) | 236 | 5 | 231 | Canonical pair-list hashes use ASCII lines \(a\): The five standard amicable pairs with smaller member at most \(10^8\) but larger member above \(10^8\) are: The checker independently trial-factors all ten endpoints. Consequently, te Riele's number 236 (smaller member bounded) and this report's number 236 (both members bounded, but diagonals allowed) are numerically equal for different reasons: For \(n\geq2\), set If all three are prime, define This gives an amicable pair. Here is the from-scratch proof, (a) elementary-rigorous. The factors in (6.2) are coprime and Therefore Direct expansion gives so The checker trial-proves the required small primalities and re-factors the resulting pairs for \(n=2,4,7\): The exact missing lemma for obtaining infinitude from this route is: > There are infinitely many \(n\geq2\) for which all three integers in > (6.1) are prime. No audited source proves this. Finite searches cannot replace it. There is also a rigorous quantitative obstruction to using this family for the proposed strong lower bound. Since the number of possible Thābit exponents with \(b_n\leq x\) is \(O(\log x)\), even before primality is imposed. Thus even a hypothetical proof that every candidate worked would give only logarithmically many pairs. Euler's generalizations and breeder constructions still impose simultaneous primality conditions; the audited literature supplies construction identities and upper-bound machinery, not a uniform lower-bound theorem producing \(x^{1-o(1)}\) actual pairs. The last sentence is (c) plausible/structural-unverified as a diagnosis of the broader machinery, not a no-go theorem. The standalone verifier uses only the Python standard library: Observed on this VM: No computation exceeding a few CPU-minutes was run. Extending a finite census cannot establish infinitude or the asymptotic lower bound. The remaining obstacle is a uniform existence theorem—already a simultaneous-prime theorem for the narrow construction (6.1), and substantially more than that for \(x^{1-o(1)}\)—not a missing finite range. PARTIAL: exact equation-only A(10^8)=236 (231 distinct pairs plus 5 perfect diagonals), with a dual-engine checker through 10^7, endpoint re-factorization through 10^8, an exact aliquot-cycle reduction, and a precise simultaneous-primality wall; infinitude and the x^{1-o(1)} lower bound remain open.2. Definition issue: the live text versus standard usage
3. Literature and claim audit
3.1 Primary sources actually checked
3.2 A citation discrepancy on the live page
3.3 An external “proof” claim that does not pass verification
4. Why the finite enumeration is complete
4.1 Factorization engine
4.2 Independent divisor-addition engine
5. Exact data
a,b\n, ordered by increasing
cfb0ae6146ce16eb90f2234a57f8b3297c82b933c24aad78afc67c9783cfd8ae;
e3701008bb918146d35f916b239362e48ee86471dd573f6d4fc0ef0fc3d810c6.6. The standard construction and the exact missing lemma
7. Reproduction and limits
# Fast regression run
python3 runs/erdos830_wave7o_reverify.py --limit 1000000 --engine both
# Full independent-engine comparison used in this report
python3 runs/erdos830_wave7o_reverify.py --limit 10000000 --engine both
# Exact 10^8 census; add --print-pairs for the complete stream
python3 runs/erdos830_wave7o_reverify.py --limit 100000000 --engine factor