Cyclic number (group theory)

A cyclic number^[1]^[2] 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.^[3]

Any prime number is clearly cyclic. All cyclic numbers are square-free.^[4] 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, ... (sequence A003277 in the OEIS).

References

^ Pakianathan, J.; Shankar, K. "Nilpotent Numbers" (PDF). Amer. Math. Monthly. 107 (7): 631–634. doi:10.2307/2589118. Retrieved 21 May 2021.
^ Carmichael Multiples of Odd Cyclic Numbers
^ See T. Szele, Über die endlichen Ordnungszahlen zu denen nur eine Gruppe gehört, Com- menj. Math. Helv., 20 (1947), 265–67.
^ 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).

[1] Pakianathan, J.; Shankar, K. "Nilpotent Numbers" (PDF). Amer. Math. Monthly. 107 (7): 631–634. doi:10.2307/2589118. Retrieved 21 May 2021.

[2] Carmichael Multiples of Odd Cyclic Numbers

[3] See T. Szele, Über die endlichen Ordnungszahlen zu denen nur eine Gruppe gehört, Com- menj. Math. Helv., 20 (1947), 265–67.

[4] 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).

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