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, 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, ... (sequence A003277 in the OEIS).

References[edit]

  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).