This is a mathematical problem, 3Q ~ (≥ ▽≤)/~ There are 400 pages in a book. The page numbers are 1, 2, 3, 4.398 and 399. How many numbers are there in the number of pages?

This is a mathematical problem, 3Q ~ (≥ ▽≤)/~ There are 400 pages in a book. The page numbers are 1, 2, 3, 4.398 and 399. How many numbers are there in the number of pages?

There are 19 "5" in 1-99
There are 19 "5" in 100-199
There are 19 "5" in 300-400
19 + 19 + 19 = 54 "5"

Find all prime numbers P such that {2 ^ (p-1) - 1} / P is a complete square number

Consider which positive integers n, 2 ^ n - 1 or 2 ^ n + 1 are perfect squares
(1) For the former, when n = 1, 2 ^ n - 1 = 1 is a complete square
When n ≥ 2, 2 ^ n-1 ≡ 3 (MOD 4), so it cannot be a complete square
(2) For the latter, if x ^ 2 = 2 ^ n + 1, then 2 ^ n = x ^ 2-1 = (x + 1) (x-1)
Because at least one of X + 1 and X-1 is not divisible by 4, the other must be divisible by 2 ^ (n-1)
From this we can get x ≥ 2 ^ (n-1) - 1,2 ^ n = x ^ 2-1 ≥ (2 ^ (n-1) - 2) · 2 ^ (n-1)
2 ^ (n-1) ≤ 4, so n ≤ 3
For n = 1, 2 and 3, we know that 2 ^ n + 1 = 9 is a complete square only when n = 3
Go back to the original topic
For prime number P > 2, let P = 2K + 1 and K be a positive integer
So (2 ^ (p-1) - 1) / P = (2 ^ (2k) - 1) / P = (2 ^ k-1) (2 ^ k + 1) / P
Note that (2 ^ k-1,2 ^ k + 1) = (2 ^ k-1,2) = 1, that is, 2 ^ k + 1 and 2 ^ k-1 are coprime
The prime number P | (2 ^ k-1) (2 ^ k + 1), so p divides one of them
(1) If P | 2 ^ k-1, then (2 ^ k-1) / P is an integer
On the one hand, 2 ^ k + 1 and (2 ^ k-1) / P are also coprime,
On the other hand, the product of the two is a perfect square
This shows that both are perfect squares
It has been proved that 2 ^ k + 1 is a complete square number only when k = 3
The corresponding P = 7 can be verified to meet the requirements
(2) If P | 2 ^ k + 1, then (2 ^ k + 1) / P is an integer
It is also coprime by 2 ^ k-1 and (2 ^ k + 1) / P, and their product is a complete square,
It can be concluded that both are perfect squares
It has been proved that 2 ^ k-1 is a complete square number only when k = 1
The corresponding P = 3 can be verified to meet the requirements
To sum up, the prime numbers P are only 3 and 7