Claims

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1975_01_01_erdos_straus: Erdős and Straus's 1975 solution to Problem 387 of Nieuw Archief: every non-constant sign sequence on the positive integers has a finite zero-sum of signed reciprocals, the case A = N of the first question.

1975_01_01_sattler: Sattler's 1975 solution to Problem 387 of Nieuw Archief: every non-constant sign sequence on the denominators 2i+1 has a finite zero-sum of signed reciprocals, the case of the odd numbers in the first question.

1982_01_01_erdos: Erdős's observation, published in Sattler's 1982 squarefree-numbers paper, that an infinite set with exactly one even element has a sign pattern with no zero-sum of signed reciprocals; the second question is answered no.

1982_01_01_sattler: Sattler's 1982 theorem that every infinite arithmetic progression has property P1, so that each non-constant sign assignment on it has a finite subset whose signed reciprocals sum to zero; the first question, answered yes.

2026_02_01_larsen: Larsen's Theorem 6, that every two-part partition of the perfect squares greater than one has finite subsets of both parts with equal reciprocal sums; the squares other than 1 have property P1, the third question yes.

2026_08_16_alexeev: A Lean development in Alexeev's repository, with Codex and GPT-5.6 Sol as its formal authors, proves that every infinite arithmetic progression has property P1, the first question; it names no published proof as its source.

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