Paper Citation Record · LEDGER

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers

As of 11 August 2026, this Paper Citation Record lists 16 of 16 outbound references and 1 inbound Pith citation observation for arXiv:2607.10431.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2607.10431 v1

Coverage vector

measured 16 of 16 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-07-14T11:47:12.385640Z

measured 17 of 17 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-11T06:34:44.6726+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-01T15:54:48.081878Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

16 of 16 outbound references displayed

verified exact0
verified fuzzy0
unresolved16
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 48f22f49-44d1-4926-b0b1-739bc2d58d5d · outbound

This paper cites Bloom,Erdős Problem #650,https://www.erdosproblems.com/650; accessed July 11, 2026.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Bloom,Erdős Problem #650,https://www.erdosproblems.com/650; accessed July 11, 2026

Reference 1

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e3761677-1732-4462-b6a6-d71a22259201 · outbound

This paper cites an unresolved cited work.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Unresolved cited work

Reference 2

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation daf8f85c-3486-42e5-80f7-33f470cef095 · outbound

This paper cites an unresolved cited work.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Unresolved cited work

Reference 3

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 0d7584f3-c440-4c2e-bbfc-30e2fc30c368 · outbound

This paper cites an unresolved cited work.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Unresolved cited work

Reference 4

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 302a1680-6cbb-40ce-b3e7-a56df689b1b4 · outbound

This paper cites an unresolved cited work.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Unresolved cited work

Reference 5

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d5b1d923-82a8-4683-aad3-2f73d52c11fb · outbound

This paper cites an unresolved cited work.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Unresolved cited work

Reference 6

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ad01e2e3-9ecd-4817-b788-60dab16a3b15 · outbound

This paper cites 2, 147–161.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers 2, 147–161

Reference 7

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f0695076-5e9a-483a-a461-5de5b716f509 · outbound

This paper cites Hall,On representatives of subsets, Journal of the London Mathematical Society10(1935), no.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Hall,On representatives of subsets, Journal of the London Mathematical Society10(1935), no

Reference 8

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4fbde8da-027b-4daf-8088-f501b64fa853 · outbound

This paper cites 1, 265–290.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers 1, 265–290

Reference 9

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c63ec7f1-59e1-4122-b131-05dea3eeb866 · outbound

This paper cites 2, 411–484.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers 2, 411–484

Reference 10

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4405a367-fa58-4391-b906-7646c74cc1d2 · outbound

This paper cites Ram Murty,Grimm’s conjecture and smooth numbers, Michigan Mathematical Journal61(2012), no.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Ram Murty,Grimm’s conjecture and smooth numbers, Michigan Mathematical Journal61(2012), no

Reference 11

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d0ae530c-e02b-436d-bb08-3e6d655b5012 · outbound

This paper cites Ramachandra, T.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Ramachandra, T

Reference 12

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9ad293eb-8d50-4a82-96ad-66fe0b28cbeb · outbound

This paper cites erdosproblems.com/forum/thread/711, September 29, 2025; accessed July 11, 2026.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers erdosproblems.com/forum/thread/711, September 29, 2025; accessed July 11, 2026

Reference 13

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6995712c-33ad-41b2-9e8d-096cb2f8055a · outbound

This paper cites 163, American Mathematical Society, 2015.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers 163, American Mathematical Society, 2015

Reference 14

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c7cbce91-a4bc-48cb-9bf6-c8bf5f5f859c · outbound

This paper cites an unresolved cited work.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Unresolved cited work

Reference 15

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f95ec64e-a4cd-44da-b4b0-3890c77fcc8c · outbound

This paper cites an unresolved cited work.

Long Intervals Without Distinct Multiples of the First $n$ Positive Integers Unresolved cited work

Reference 16

Resolution: unresolved; no resolver link, observed 2026-07-14T11:47:12.385640Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

Observation 45fdbf2b-300d-47f4-9db9-09561d330baf · inbound

Improved Bounds for Distinct Multiples in Intervals cites this paper.

Improved Bounds for Distinct Multiples in Intervals Long Intervals Without Distinct Multiples of the First $n$ Positive Integers

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-01T15:54:48.081878Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.