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?
[Math] Is it widely-known that “every odd number can be expressed as a sum of consecutive odd numbers except the odd primes”
parityprime factorizationprime numberssummation
Best Answer
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 $n$.