#### English Title

Speculations about certain outstanding properties of numbers

#### Enestrom Number

564

#### Fuss Index

39

#### Original Language

Latin

#### Published as

Journal article

#### Published Date

1784

#### Written Date

1775

#### Content Summary

Euler writes 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 Ï *D* as the number of integers less than *D* and relatively prime to *D*. He provides a table of Ï *D* up to *D*=100, then makes a table listing 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, of course, involves Ï *N*.

#### Original Source Citation

Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 1780: II, pp. 18-30.

#### Opera Omnia Citation

Series 1, Volume 4, pp.105-115.

#### Record Created

2018-09-25