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4

See this oeis entry. This can be easily reduced to Ramanujan-Nagell problem. It was conjectured by Ramanujan in 1913 and resolved by Nagell in 1948 (where the year 1930 come from is not clear).


4

Plato does not define perfect numbers anywhere. The earliest extant definition is in Euclid VII, def. 22. Although Plato uses the term "perfect number" I do not think it certain that he understood this phrase in the same way as Euclid.


3

I think that this is what you are asking for: Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques, et sur l’usage de cette transformation dans la résolution de différents problèmes, Journal de l'École polytechnique, $18^e$ cahier, 11 (1820), p. 417–489. Here you can find the scan of the original paper (in French, of course)...


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