Conjecture: 29 is largest integer <> h-gonal(i) + j-gonal(k), h, i, j, k >2

[Jim Nastos]

On 12/25/07, Jonathan Post <jvospost3 at gmail.com> wrote:

> Integers which cannot be written as the sum of two polygonal numbers
> of indices > 2:
> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 23, 29, ...|
>
> In a few minute's inspection, it seems to me that the number of ways
> of representing an integer in our sum set grows (irregularly, to be
> sure) fast enough that 29 may be the last such non-represented number.

You only need your first three rows to make all the rest of the sums.

- Any multiple of 3 larger than 9 can be represented as a sum of two
numbers in your first row.

- Any number in the form 3k+1 larger than 13 can be represented as a
sum of 10 + a number in your first row.

- The numbers larger than 13 of the form 3k+2 are:
32, 35, 38, 41, 44, 47, 50, 53, ...

- We break these into two cases:
6m + 2 = 32, 38, 44, 50, ...
6n + 5 = 35, 41, 47, 53, ...

- The ones of the form 6m+2 can be written as 10 (from 2nd row) +
another number in the 2nd row.

- The ones of the form 6n+5 can be written as 25 (from 3rd row) +
another number from the second row.

  So you don't need all the numbers in your first 3 rows to make the
rest of the sums. Just all the numbers in your first 2 rows and only
25 from your 3rd row.

  JN