Little math problem(s) February 2019

[Albert Chan]

(02-23-2019 01:23 PM)pier4r Wrote:  One has a set of positive integers
I = { i1,...., iN }

If ... meant any positive integers, with no relation to each other, S1 and S2 is not enough.

If ... meant I = {i1, i1 + 1, ... i1 + N-1}, I may be recovered from S1, S2.

S1 = N * (i1 + i1 + N-1) / 2
i1 = 1/2 - N/2 + S1/N

If N is a fixed, i1 is unique, thus I is unique.

If N is variable, substitute i1 into S2 expression, it created a quartic polynomial:

N^4 - N^2 - 12 S2 N + 12 S1² = 0

With i1 and N only allowed positive integer, my guess is I is still unique.
If 2 list have the same S1, S2 of shorter list should be bigger.

Example:
sum(k, k, 100, 200) => 15150
sum(k, k, 102, 201) => 15150
sum(k^2, k, 100, 200) => 2358350
sum(k^2, k, 102, 201) => 2378550