The Consecutive Integer Partition Theorem, also known as the Odd Divisor Proprety of Consecutive Sums or a proprety of Sylvester’s Theorem, states that: for any positive integer N, the total number of ways to express N as the sum of one or more consecutive positive integers (including the trivial one-term case where the sum is N itself) is equal to the number of positive odd divisors of N.
This by itself is difficult to understand. I must personally admit that trying to comprehend this monstrous statement is equilivant to swallowing an American bullfrog.
You are given any positive integer N. Without loss of generality, assume it is 12.
How many odd factors does 12 have? They are 1 and 3. So 12 has 2 odd factors.
Recall that this equals to “the total number of ways to express N as the sum of one or more consecutive positive integers.” So, the total number of ways to express n as {the sum of consecutive positive integers} is 2.
For example, we can express 12 as 3 + 4 + 5. This is indeed a sum of consecutive positive integers. 12 is also equal to 12 (Recall that 12 is also a sum of one or more consecutive positive integers.) There aren’t any more ways (see for yourself!)
So yes, it is true that given a positive integer N, its total number of odd factors equals the number of ways to write N as the sum of one or more consecutive positive integers.
One more example:
14. 14 has 2 odd factors, 1 and 7. We can write 2 + 3 + 4 + 5 and just 14 by itself. This equals two ways so yes, the Consecutive Integer Partition Theorem (for these two cases, at least) is true!
Don’t Forget! When finding factors of N, don’t forget that 1 also qualifies as an odd number.
Also Don’t Forget! When finding ways to express N as a sum of some numbers, don’t forget that N itself is also a way. For example, 86 is also a way to express 86 as the sum of ONE (or more) consecutive positive integers.
