Given positive integer n, consider the set of numbers {n²+1, n²+2, ... (n+1)²}. If we pick two numbers x and y out of that set, how many different values can the product xy take?
(In reply to
re: Where's the proof? by Josh70679)
I should flesh out the claim I made that ix > n2 implies i + x > 2n. Let k be a positive integer < n, and WOLOG let i = n - k. If ix > n2, then
- x > n2/(n-k)
- = n(1 + k/(n-k))
- = n + nk/(n-k)
- = n + k(1 + k/(n-k))
- = n + k + k2/(n-k)
- > n + k
So i + x > n-k + n+k = 2n.
re(2): Where's the proof?
