Question

Determine the greatest common divisor of the elements of the set \[ s = \{ n^{13} - n \mid n \in \mathbb{z} \}. \]

Answer 1

2730

Explanation:

We want to determine the greatest common divisor of the elements of the set
`S = \{ n^(13) - n \mid n \in \mathbb{Z} \}.`

We apply the Fermat's little theorem which states that if p is a prime number, then for any integer a, the number aᵖ − a is an integer multiple of p.

Now,
`n^(13) \equiv n \mod p` if p-1 divides 12.

Since the of 12 are 1,2,3,4, 6, 12, the corresponding primes are 2, 3, 5, 7, 13.

Therefore, the gcd of the elements in
`2^(13)-2` and
`3^(13)-3$ is 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13.`

2*3*5*7*13=2730

Therefore, the gcd of the elements in set S is 2730.