Cyclic number (group theory)

{{Short description|Number n where n and totient(n) are coprime}}

A cyclic number{{cite journal |last1=Pakianathan |first1=J. |last2=Shankar |first2=K. |date= |title=Nilpotent Numbers |url=http://www.math.ou.edu/~shankar/papers/nil2.pdf |journal=Amer. Math. Monthly |volume=107 |issue=7 |pages=631-634 |doi=10.2307/2589118 |access-date=21 May 2021}}[http://www.numericana.com/data/crump.htm Carmichael Multiples of Odd Cyclic Numbers] is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic.See T. Szele, Über die endlichen Ordnungszahlen zu denen nur eine Gruppe gehört, Com- menj. Math. Helv., 20 (1947), 265–67.

Any prime number is clearly cyclic. All cyclic numbers are square-free.For if some prime square p² divides n, then from the formula for φ it is clear that p is a common divisor of n and φ(n).

Let n = p₁ p₂ … p_k where the p_i are distinct primes, then φ(n) = (p₁ − 1)(p₂ − 1)...(p_k – 1). If no p_i divides any (p_j – 1), then n and φ(n) have no common (prime) divisor, and n is cyclic.

The first cyclic numbers are 1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, ... {{OEIS|A003277}}.