Is it widely-known that "every odd number can be expressed as a sum of consecutive odd numbers except the odd primes"?

Question

I've been working on the Collatz conjecture and stumbled across what I think is a major property of prime numbers: every odd number can be expressed as a sum of consecutive odd numbers except the odd primes.

Is this a new discovery?

Answer 1

This is an interesting observation - to answer your question, yes, it's well known.

First I will try to clarify what I think you mean.

You can always express an odd number as a sum of one consecutive odd number - itself. I assume this is a trivial case you want to ignore, so your "consecutive" means "more than one".

Then you are right. Primes (and $1$ itself) are the only odd numbers that can't be expressed as a sum of two or more consecutive odd numbers.

You can put together a proof this way:

The sum of the first $n$ odd integers is $n^2$. That means the sum of consecutive odd integers is a difference of squares $m^2-n^2 = (m-n)(m+n)$. Then given a composite odd number $ab$ with $a$ and $b$ odd and greater than $1$ you solve $a = m+n$ and $b = m-n$ to find $m$ and $n$.

Answer 2

Any sum of consecutive odd numbers can also be expressed as a product of the average (which will be an integer) and the count of values. Therefore if we confine the odd numbers to positive values and the count to be $\ge 2$, the result cannot be a prime number.

For example, $$7+9+11+13+15 = \overset{\text{average}}{11}\times\overset{\text{count}}5 = 55$$

If we allow negative odd numbers into the sum, then we can arrange for the average to be $1$ and any odd number is possible. We can also make any number divisible by $4$, but not even numbers which are twice an odd number.