In this chapter we introduce the concepts of order and the primitive root, two of the more fascinating and useful ideas in number theory. On the fundamental side, they helped the young Gauss to reduce the equation x 16 + x 15 + ... + x + 1 = 0 to several quadratic equations leading to the construction of the regular 17-gon. These same concepts also allow us to see why the decimal fraction of 1/7 has a period of length 6, while the decimal fraction for 1/11 has a period of only 2. And why does 1/99007599, written as a binary fraction, have a period of nearly 50 million 0’s and l’s? We shall see!
KeywordsPeriod Length Primitive Root Great Common Divisor Decimal Fraction Coprime Factor
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