Let σ(n) be the sum of the positive divisors of an integer n and ϕ(n) be the number of positive integers smaller than n that are coprime to n. If p is a prime number, what is the maximum value σ(p)/ϕ(p)?
To find the maximum value of σ(p)/ϕ(p), we need to analyze the expressions for σ(n) and ϕ(n) when n is a prime number.
For any prime number p, σ(p) represents the sum of all divisors of p. Since p is a prime number, it only has two divisors: 1 and p. Therefore, σ(p) = 1 + p = p + 1.
On the other hand, ϕ(p) represents the number of positive integers smaller than p that are coprime to p. Since p is a prime number, the only positive integers smaller than p are 1, 2, 3, ..., p-1. For any number in this range to be coprime to p, it must not have any factors in common with p other than 1. Since p is prime, no number in this range would have any factors in common with p. Therefore, ϕ(p) = p-1.
Now, we can determine the maximum value of σ(p)/ϕ(p) by substituting the above values:
σ(p)/ϕ(p) = (p + 1)/(p - 1)
To maximize this expression, we need to find the maximum value of (p + 1)/(p - 1) for any prime number p.
To do that, we can simplify the expression:
(p + 1)/(p - 1) = (p - 1 + 2)/(p - 1) = 1 + 2/(p - 1)
Since p is a prime number, p - 1 is always greater than 1. Therefore, the fraction 2/(p - 1) is always less than or equal to 2.
Thus, the maximum value of σ(p)/ϕ(p) = 1 + 2/(p - 1) is achieved when the fraction 2/(p - 1) is equal to its maximum value of 2.
Therefore, the maximum value of σ(p)/ϕ(p) is 1 + 2 = 3.
So, regardless of the value of p (as long as it is a prime number), the maximum value of σ(p)/ϕ(p) is 3.