E564 -- Speculationes circa quasdam insignes proprietates numerorum

(Speculations about certain outstanding properties of numbers)


Summary:

Euler says that "without a doubt, the number of all the different fractions between the endpoints 0 and 1 is infinite; and since the number of all the integers is also infinite, it is manfiest that the multitude of all the ordinary fractions up to infinity is greater; and at the same time, there must be innumerably many different fractions between any two numbers that differ by one." He goes on to define p(D) to be the number of integers less than I > D and relatively prime to D. He provides a table of p(D) up to D = 100, then makes a table of the number of fractions with denominators less than or equal to n, for n = 10, 20, 100 and poses the problem of finding this number of fractions for any given number N. The solution involves p(N).

According to the records, it was presented to the St. Petersburg Academy on October 9, 1775.

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