Abstract
After defending in general terms the value of arithmetic demonstrations, where, according to Euler, the force of genius shines more brightly than in any other kind of demonstration, he goes on to let the force of his own genius shine forth in demonstrating the fact that, for any pair of relatively prime numbers a and N, aϕ(N) − 1 is always divisible by N, where we anachronistically denote by ϕ(N) the number of numbers less than N that are relatively prime to it. He approaches this theorem by considering the remainders that result when the numbers in an arithmetic or geometric sequence are each divided by any number n. By carefully considering the remainders of arithmetic sequences, he is able to give the general formula for the number of numbers prime to a given number, a formula based on the prime factorization of the given number. In considering the remainders of geometric sequences, he is able to show that if a and N are relatively prime, then the number of remainders that occur when powers of a are divided by N divides ϕ(N). In the course of his arguments, Euler says that all numbers that have the same remainder when divided by N can be counted as identical to that remainder. He thereby comes close to treating these remainders not as integers, but as elements in the multiplicative group of congruence classes modulo N of numbers that are prime to N.
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Recommended Citation
Lasell, Brendon
(2026)
"Arithmetic Theorems, Demonstrated by a New Method: an English Translation of E271,"
Euleriana: 6(2), pp.138-160.
DOI: https://doi.org/10.56031/2693-9908.1114
Available at:
https://scholarlycommons.pacific.edu/euleriana/vol6/iss2/3