Cyclic number (group theory)

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A cyclic number[1] 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 iff any group of order n is cyclic.[2]

Any prime number is clearly cyclic. All cyclic numbers are square-free.[3] Let n = p1 p2pk where the pi are distinct primes, then φ(n) = (p1 − 1)(p2 − 1)...(pk – 1). If no pi divides any (pj – 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, ... (sequence A003277 in OEIS).


  1. ^ Carmichael Multiples of Odd Cyclic Numbers
  2. ^ See T. Szele, Über die endlichen Ordnungszahlen zu denen nur eine Gruppe gehört, Com- menj. Math. Helv., 20 (1947), 265–67.
  3. ^ 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).