nontotient

A nontotient is a positive integer that is not a totient number. A totient number, also known as a φ-number, is a positive integer that can be expressed as the Euler's totient function of some positive integer. The Euler's totient function, denoted as φ(n), counts the number of positive integers up to a given integer n that are relatively prime to n.

The concept of nontotients arises from the study of the properties of the Euler's totient function and its relationship with other mathematical functions. While totient numbers are those that can be expressed as φ(n) for some n, nontotients are those that cannot. For example, the number 1 is a totient number because φ(1) = 1, but the number 2 is a nontotient because there is no positive integer n such that φ(n) = 2.

The study of nontotients is a relatively new area of research in number theory, and much remains to be understood about their properties and distribution. However, it is known that nontotients are relatively rare compared to totient numbers, and they exhibit interesting patterns in their distribution.

In summary, a nontotient is a positive integer that cannot be expressed as the Euler's totient function of any positive integer. The study of nontotients is an active area of research in number theory, and much remains to be understood about their properties and distribution.