I am trying to solve this problem , but it is difficult for me to understand this:
Let φ be the Euler function function, i.e. for a natural number n, φ (n) is the number k, 1 <= k <= n, for which gcd (k, n) = 1.
By iteration φ, each positive integer generates a decreasing chain of numbers ending in 1. For example. if we start with 5, a sequence of 5,4,2,1 is generated. Here is a list of all 4 chains:
5,4,2,1 7,6,2,1 8,4,2,1 9,6,2,1 10,4,2,1 12,4,2,1 14,6,2,1 18,6,2,1
Only two of these chains begin with prime; their sum is 12.
What is the sum of all primes less than 40,000,000 that generate a chain of length 25?
My understanding of this is that φ (5) is 4, 2, 1 - i.e. reciprocal numbers up to 5 are 4, 2 and 1 - but then why is there no 3 in this list either? As for 8, I would say that 4 and 2 do not coincide with 8 ...
I think I must have misunderstood the question ...
Assuming that the question is poorly formulated, and φ (5) is 4, 3, 2, 1 as a chain of 4. I do not find a single prime number that is less than 40 m that generate a chain of 25 - I find chains of 24, but they relate to non-empty numbers.