Primitive Root

I’m surprised that I haven’t come across this concept before but it was brought to my attention via one of the properties of 26701, my diurnal age on Wednesday, May 11th 2022. It is a member of OEIS A061334: primes with 22 as smallest positive primitive root. The initial members of this sequence are:

3361, 6841, 9439, 13681, 14449, 26591, 26701, 28729, 39373, 40609, 41161, 41521, 

54601, 61031, 66071, 66301, 68041, 68881, 70729, 82021, 85201, 89209, 90217, 93601, 

96769, 104831, 110161, 112921, 117721, 121631, 125329, 126001, 128521

The question that I naturally asked was: what is a primitive root?

This YouTube video provides a good explanation and Figure 1 shows a screenshot of the author’s simplified definition:

Figure 1

Using this definition, it’s easy enough to show that 22 is the smallest primitive root of 26701 (permalink plus see SageMathCell verification below).

Some other larger primitive roots are 26, 29, 38, 47, 59 and 66. In fact, as the calculation above shows, there are a total of \( \phi (\phi(26701)=7040 \) primitive roots where \( \phi \) is the euler totient function. This YouTube link to Michael Penn’s video explains why this is so. What is less apparent however, is the usefulness of finding the primitive root of a prime number. I should investigate this further at some point but I’m still on holidays and my mind in holiday mode.