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A New Characteristic of the Identity Function

✍ Scribed by Jean-Marie De Koninck; Imre Kátai; Bui Minh Phong

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
338 KB
Volume
63
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

In 1992, C. Spiro [7] showed that if f is a multiplicative function such that f (1)=1 and such that f ( p+q)= f ( p)+ f (q) for all primes p and q, then f(n)=n for all integers n 1. Here we prove the following:

for all primes p and integers m 1, (1) then f (n)=n for all integers n 1.

Proof. First we show that

( 2 ) Indeed, using (1) and the fact that f is multiplicative, we have

from which (2) follows immediately. We now show that f (n)=n for all positive integers n 12.

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