evidence boundary
supportsfrontiers / frontier
Erdős problems frontier
- id
- vfr_37aec80d874a0239
- license
- CC-BY-4.0
- findings
- 1,256
- accepted core
- 6
- contested
- 0
- links
- 17
- sources
- 1,234
- evidence
- 1,256
- avg conf
- 0.98
e1288/1288 · statement.registered · agent:claude-proxy · 2026-06-10 · null→null
Evidence atom
back to sources{"artifact_id":"va_9bc926d75e4e3881","artifact_packet_id":"cap_61973ee16b553d57","candidate_claim_id":"vc_f4d6defc70020a0c"}
- id
- vea_d80bf9580adb1280
- frontier
- Erdős problems frontier
- source
- vs_a84f6f2d0c5bfab6
- finding
- vf_a566be23717e9a21
finding binding
boundopen_question
Erdős Problem #1051 has status 'proved (lean)'. Statement: Is it true that if $a_0 < a_1 < a_2 < \cdots$ is a strictly increasing sequence of integers with $\liminf a_n^{1/2^n} > 1$, then the series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$ is irrational? This was solved in the affirmative by Aletheia [Fe26]. This was extended by Barreto, Kang, Kim, Kovač, and Zhang [BKKKZ26], who essentially give a complete answer: if $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio and $1\leq a_1 < a_2 < \cdots$ is a monotonically increasing sequence of integers such that $\limsup a_n^{1/\phi^{n}}=\infty$ then $\sum_{n=1}^\infty \frac{1}{a_na_{n+1}}$ is irrational. Conversely, for any $1 < C < \infty$ there exists a sequence of integers $1\leq a_1<\cdots$ such that $\lim a_n^{1/\phi^{n}}=C$ where this infinite sum is a rational number. (Further, more general, results are available in [BKKKZ26].) This was formalized in Lean by Baretto. Topics: irrationality. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.
source binding
source-boundcap_61973ee16b553d57 · vc_f4d6defc70020a0c
vs_a84f6f2d0c5bfab6
review context
unverified1 events
1 reviewable changes and 0 evaluation records target this atom or its bound objects.
statement
{"artifact_id":"va_9bc926d75e4e3881","artifact_packet_id":"cap_61973ee16b553d57","candidate_claim_id":"vc_f4d6defc70020a0c"}
locator
span:0
extraction method
artifact_to_state_import
support relation
supports
condition refs
vcnd_2420a1ce9c17a528
caveats
No caveats recorded.
Review, event, and evaluation records
2events
vev_e906e3267e9339b7finding.assertedCandidate claim vc_f4d6defc70020a0c imported from artifact packet cap_61973ee16b553d57
reviewer:erdos-db-trust · 2026-05-30
reviewable changes
vpr_ef2a2364e670ce45finding.addCandidate claim vc_f4d6defc70020a0c imported from artifact packet cap_61973ee16b553d57
applied · agent:erdos-spine-ingest · 2026-05-30
evaluations
No evaluation rows are attached.