Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-05-10T18:17:05.380100Z
Paper Citation Record · LEDGER
As of 12 August 2026, this Paper Citation Record lists 68 of 68 outbound references and 0 inbound Pith citation observations for arXiv:2604.09730.
A citation records a reference. It does not transfer a finding from one paper to another.
Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-05-10T18:17:05.380100Z
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-12T06:34:41.77262+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links
A source-named dated measurement, never combined with another source.
Source: cited_works
68 of 68 outbound references displayed
External citation measurements
No source-named external measurement is stored.
Observation 9ded7cd5-5253-449f-ae1e-1b0a009ab2b9 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Alzer and F
Reference 1
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation de0439ea-dd77-4891-b531-d22295e9d2df · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Baker: A sharpening of the bounds for linear forms in logarithms
Reference 2
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation c029f850-2828-49ed-8217-e70bb20c4c7e · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Bachman, Introduction of p -adic numbers and valuation theory, Academic Press, New York
Reference 3
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 16a4cdd9-37d2-4b73-8c0c-ca6623ea5e66 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Baczkowski, M
Reference 4
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 515d1b75-59fe-4109-b9ad-cf55b1afccfe · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Baczkowski and S
Reference 5
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 0d67f955-2e7a-431b-99f3-71c68b71e915 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Baker, Experiments on the abc-conjecture
Reference 6
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 2193c45d-2c50-4b9e-8af9-17b6e7a1286c · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Unresolved cited work
Reference 7
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7693a354-6abc-4e89-b051-230e0b33a577 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Bennett, et al., Explicit bounds for primes in arithmetic progressions
Reference 8
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation c04980ef-8577-4276-a532-f2a8c8f4b324 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Unresolved cited work
Reference 9
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 52234963-f146-4993-ade6-df2f1605e200 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Bhargava, P-orderings and polynimial functions on arbitrary subsets of Dedekind rings
Reference 10
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation fc1c8efc-3998-46cc-9500-3a2ce188c2dd · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Berend and C.F
Reference 11
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation a9095cf2-6da5-4b50-93d3-422d949d5205 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Berend and J.E
Reference 12
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 08ff981e-1e82-4027-b1a3-c115266431ec · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Berndt and W.F
Reference 13
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation c8c50fcc-738d-4fe4-b624-298eeadaa0db · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Brocard: Question 1532
Reference 14
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation ab7fa1cb-945f-46e6-b11e-5a05aa227d17 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Dusart, Autour de la fonction qui compte le nombre de nombre premiers
Reference 15
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 009e6a85-3e22-4c2a-8975-dba09be6e1f6 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Dusart, In\'egalit\'es explicites pour (X), (X), (X) et les nombres premiers
Reference 16
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 6e510bf0-f57f-4abb-b278-f0d969928dcd · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Erd o s and R
Reference 17
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation d52998c1-606e-4376-be20-6447662b926b · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Erd o s, On consecutive integers
Reference 18
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 1c8787b8-4ed2-4295-86c3-9896abf0fde3 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Erd o s\ et al., On the number of divisors of n! , in Analytic number theory, Vol
Reference 19
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation ac537e49-119a-4d5b-b57b-49a14d7a7303 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Bugeaud, M
Reference 20
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 3346a683-360a-40a2-a9bf-ff4b7219e1fd · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Unresolved cited work
Reference 21
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 33eb0896-efc0-499c-afee-83d6585d88c8 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Dabrowski, On the equation n!+A=y^2
Reference 22
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 9a8bfecd-dca6-41a5-a17b-bfefbd93c587 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Dabrowski: On the Brocard--Ramanujan problem and generalizations
Reference 23
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 67e46cfc-b4d9-453c-8c2c-22d5dff03679 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Dabrowski and M
Reference 24
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 60aafbac-9b30-47dc-9095-c2d86b2f4356 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Epstein and J
Reference 25
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation d3342bf4-049a-422f-b623-af8cc0e86a17 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Dufour and O
Reference 26
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7a2522f7-ca49-4e69-bdcf-c44cfb4a5c69 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Unresolved cited work
Reference 27
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation c95e9ebe-1756-4dbe-a885-850167faebb9 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Hardy and E.M
Reference 28
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 4078d81a-8613-48f8-b35a-1fa2e64cac37 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Janusz, Algebraic number fields, Academic Press, New York and London (1973)
Reference 29
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 30a894ef-6c16-4e09-843a-3500eb0446e6 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Erd o s and R.L
Reference 30
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 88150f95-8b29-4696-be5e-58e3debe7e9c · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Kihel and F
Reference 31
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation d598a8f8-18e8-4c4f-8922-b747c6fb5b1c · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Lang: Old and new conjectured diophantine inequalities
Reference 32
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 08fbcfb9-b433-432b-8b73-a24ee42b7322 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Halberstam and H.-E
Reference 33
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 163730bd-131c-4b4a-9953-2e1180c69aa3 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Gawron, A note on the Diophantine equation P(z)=n!+m!
Reference 34
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 098573ae-7900-47bf-87da-d169575cc2a4 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Guy, Unsolved problems in number theory
Reference 35
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 5f3dc721-c61a-4296-a46f-88fb53e01d71 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Lang, Old and new conjectured Diophantine inequalities
Reference 36
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 8ee39138-4827-4671-8268-c519e1749a40 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Luca, Equations involving arithmetic functions of factorials
Reference 37
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 4f7adc7f-bd53-48b5-954e-5a9617f4fe8d · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Laishram and T.N
Reference 38
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 2b2890ff-f8db-49f2-83c4-d2340315fde9 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Luca, The Diophantine equation P(x)=n! and a result of M
Reference 39
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 0f5f891f-7fff-43ba-9647-f2abb81bf1ec · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Luca, On factorials which are products of factorials
Reference 40
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation e4b6c1ce-4615-4bd1-95fd-b6a77b4ebae4 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Unresolved cited work
Reference 41
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation e5c961a3-fa3b-4376-b3a7-637538b80af2 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Luca, On the Diophantine equations f(n)=u!+v!
Reference 42
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 19a49dc1-57ee-4e31-b90a-770ef24b32a4 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Matson, Brocard's problem 4th solution search utilizing quadratic residues
Reference 43
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation ffef8088-4d8e-4e8e-98b0-9e8994944dd6 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Makki Naciri, On the variant Q(n!)=P(x) of the Brocard--Ramanujan Diophantine equation
Reference 44
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 53180916-556d-47fc-994b-cc69e1469042 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Makki Naciri, On the Brocard-Ramanujan equation with 7-free integers and prime powers
Reference 45
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 827b7864-f12c-40e3-87c0-95b09dab8658 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Unresolved cited work
Reference 46
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation b745b8da-5188-4d10-a615-f0662f30b245 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Nair and T.N Shorey, On products from blocks of consecutive odd primes
Reference 47
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 1530f752-be66-4719-86ee-b003d1fceb2b · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Novakovi\'c, Diophantine equations involving double factorials
Reference 48
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation fff2556d-252e-4bc1-8efc-fa92b03b74cf · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Novakovi\'c, A note on some polynomial-factorial Diophantine equations
Reference 49
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 1a566883-8133-4111-a713-e7cbfa5e2be5 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Administrative Law's Fourth Settlement: AI and the Scrutable State
Reference 50
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 6d0bd9ff-9343-48cb-aec3-ce03993f74d7 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ On the diophantine equation $An!+Bm!=f(x,y)$
Reference 51
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7d31df3f-b4a9-48c6-b1b5-c9cfe0918f98 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Novakovi\'c, The Diophantine equation P(x)= r i=1 H_ n_i
Reference 52
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 27586a43-4cbf-461d-8267-4b3f10e31402 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Overholt, The Diophantine equation n!+1=m^2
Reference 53
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 92da070b-36a4-4a77-94cc-da9eb1043f62 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Pollack and H.N
Reference 54
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 2aadbf4d-2f6f-43c2-ab1a-ca561b3047fe · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Ramanujan, Question 469
Reference 55
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 99d6b2c2-c2f7-42e1-a239-df80b1f1da70 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Rosser and L
Reference 56
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 49fb0416-b208-4cb5-9ba8-307f9dd0c269 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Saunders, Diophantine equations involving the Euler totient function
Reference 57
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7e1efc6d-aed3-411b-a697-6d2efbd395df · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Shorey: Diophantine approximations, Diophantine equations, transcendence and applications
Reference 58
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 1b6f2cf1-b24b-48df-aadd-54cbb51961e6 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Shorey and R
Reference 59
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation bcee3b06-ab50-4ad0-932d-eaccafd1dde5 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Siksek: Diophantine equations after Fermat's last theorem
Reference 60
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 5ddbe455-2e31-47eb-8c8d-822c11b04a07 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Takeda, Finiteness of trivial solutions of factorial products yielding a factorial over number fields
Reference 61
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 8ffad563-e6f4-4bf8-9fe1-70a48e275889 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Takeda, On the finiteness of solutions for polynomial-factorial Diophantine equations
Reference 62
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 7c5beda0-431b-498f-95cd-2ffd0e6a60ef · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Takeda, Product of Factorials Equal Another Product of Factorials
Reference 63
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation b9fcf0d3-fdf7-4339-8d42-bf14170a801d · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Siegel: Aproximation algebraischer Zahlen
Reference 64
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 2a78e327-4989-45a4-ba4a-73967623aa03 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Tijdeman: Applications of the Gel'fond--Baker method to rational number theory
Reference 65
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 9f5388ef-4a2a-415b-ac6d-427ccabc20ac · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Ulas, Some observations on the Diophantine equation y^2=x!+A and related results
Reference 66
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 91ea14e9-3f58-4ab4-9768-a06e3b6f3ec0 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Ulas, Some experiments with Ramanujan--Nagell type Diophantine equations
Reference 67
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
Observation 32b0a895-8007-4255-9eb2-0d8570803778 · outbound
A comment on the equation $n!!=a_1!!\cdots a_t!!$ Yamada, A generalization of the Ramanujan--Nagell equation
Reference 68
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.
No inbound Pith citation observations are available.