Memorandum for friends on amicability

TDV lists al-Farisi's Memorandum for friends on amicability and identifies Rashed's1982 Journal of the History of Arabic Science edition and later study. This identifies the work through a list actually read; it does not claim that the edition or manuscript was inspected here ².

MacTutor describes its factorization/combinatorial proof approach to Thabit ibn Qurra's older rule. That rule retains its author; this is neither the first knowledge of220/284 nor the invention of all number theory. Its question differs from the Revision of the Optics, despite a shared author ³.

The biography begins with a rule requiring three expressions built from powers of2 to be prime before constructing an amicable pair. It describes al-Farisi's change as more than a minor proof revision: prime-power structure and combinatorial choices. MacTutor attributes the uniqueness-of-factorization interpretation and attempted proof to Rashed and notes disagreement over whether the proof was complete ⁴.

The reported closing examples are220/284 and17296/18416. For the latter, the source describes lemmas establishing1151's primality, including a sieve application. Our modern explanation does not reconstruct their original sequence from an unread text. Present-day divisor-sum arithmetic checks the example; it is not a critical reading of the treatise's lines ⁵.

MacTutor reports Hogendijk's argument that Thabit may already have known the larger pair. We therefore do not declare it al-Farisi's first discovery or assign it to the first familiar European name. Nor does discussion of unique factorization become a definitive priority verdict for proving it. The disagreement itself belongs before the reader ⁶.

A generating rule also does not establish that it generates every pair: sufficiency is not a complete catalogue. The museum explains this with modern examples, separating the inclusive divisor sum from the proper-divisor sum after self-subtraction. Contemporary notation and arithmetic are not claimed as copied historical tables or symbols ⁷.

The late thirteenth-/early fourteenth-century horizon is lifetime context, not a precise composition date. TDV's name and dates differ from MacTutor; the underlying student's colophon was not independently inspected. We do not select a writing day from alternative reported death dates ^{8 9}.

The actual biography and TDV list were read, not the Arabic treatise,1982 edition, or full Hogendijk paper. Ordinary publisher access stopped at human verification; the priority argument remains reported through MacTutor. These are explicit reading boundaries, not a new critical edition implied despite missing access ^{10 2}.