Sums of Consecutive Integers

Number theory problems can be some of the most difficult problems, but they are usually the most interesting and satisfying to solve. Recently, I was tasked with finding the solution to this problem: Find out which positive numbers cannot be expressed as the sum of two or more consecutive positive integers.

The first thing we can do is to try some small numbers. After trying a few, a pattern starts to emerge. All the powers of 2 seem to be the only numbers unable to be expressed as the sum of two or more consecutive positive integers. Although this is a pattern, we can only say that this pattern holds for small numbers. Now this question arises: Can we prove that this holds for all numbers, that only powers of two cannot be expressed as the sum of two or more consecutive positive integers?

Initially, I was unsure how to go about this. So I decided to find the numbers that could be expressed as a sum of two or more consecutive positive integers to help find the numbers that could not.

Since any odd number can be represented as , and , all odd numbers greater than one can be expressed as the sum of two consecutive positive integers. Let us call this statement A.

However, I could not figure out what to do after that. After thinking about it for a while, I remembered an old trick I had discovered. Any number i that is divisible by an odd number n can be represented as the sum of n consecutive integers centered around . For example, the number 40 is divisible by the odd number 5 (). Then, the trick states that 40 is equal to the sum of five integers centered around 8. Indeed, we see that The proof for this is quite simple (omitted here), but it suffices to say that it works because the sum of every pair of numbers around is equal to n. Hence, all numbers divisible by an odd number can be expressed as the sum of consecutive integers.

Combining statement A and statement B, we find that all numbers that are divisible by an odd number can be represented as the sum of two or more consecutive integers.

To find out whether a number is divisible by an odd number, we find its prime factorization. Since 2 is the only even prime number, if the prime factorization contains any number besides 2, the number is divisible by an odd number, and it can be represented as the sum of consecutive integers. The only numbers that do not fit into this category are the numbers whose prime factorizations contain only 2. In other words, the only numbers that cannot be expressed as the sum of two or more consecutive positive integers are the powers of 2 (1, 2, 4, 8, …).

This proof was the result of thinking by myself as well as fruitful discussions with my father. He was trying to solve the problem too, and at a point when we were both stuck, we sat down together and shared our separate insights, and soon after, we figured it out. This was a great example of how working together can help both parties because I would have taken much longer to solve this problem without his input (if I would have solved it at all!).