Consider you have created grid of numbers like following image starting from any positive integer (in this case 8)
To create such grid, follow this steps;
- Pick a number greater than one and write down $n, n-1, n-2, ..., 1$ as column headers
- Likewise, write down $n, n-1, n-2, ..., 1$ as row headers
- Fill each cell with $row_header * column_header$
If you group numbers diagonally starting from upper left, you get these groups of numbers
- {64}
- {56, 56}
- {48, 49, 48}
- {40, 42, 42, 40}
- etc.
Let $G_n$ denote nth group. I want to prove if following theory is correct. $$ i < k \Rightarrow \forall x \in G_i, \forall y \in G_k, x > y $$
It seems to hold for small enough starting numbers like $8$. It is trival to prove that any cell will be bigger then the cells to the right on the same row and will be bigger then the cells below it on the same column. But, I can't be sure about diagonal groups.
