English Title

Speculations about certain outstanding properties of numbers

Authors

Leonhard Euler

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 π D as the number of integers less than D and relatively prime to D. He provides a table of π 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 π 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

Included in

Mathematics Commons

Share

COinS