On Friday, August 15, 2014 11:35:11 PM UTC-5, Archimedes Plutonium wrote: > Now let me ask a question out of curiousity. Perhaps I asked this before on Goldbach. What can we do with perfect squares as sums. Here are the first several perfect-squares 1, 4, 9, 16, 25, . . and the question I ask is can they provide us with a Goldbach type of conjecture. Could we say that all numbers, both even and odd are the sum of no more than 4 perfect-squares? Say for instance 15 is the sum of 1+1+4+9 = 15. > > > > So, what can we do with perfect-squares rather than with primes? >
I think I might be onto something for I do not recall any conjecture or theorem for perfect squares constructing all the Naturals. Now suppose 4 perfect-squares can construct all the naturals, then, what would be a conjecture of how many primes construct all Naturals beyond 5? It would be nice if we could take 1 as prime and then we could say that all Naturals are the sum of no more than 3 primes by using Goldbach and 1.
Now, given only even numbers we cannot get any odd numbers, but given only odd numbers we can get all Naturals beyond 0 by 2 odd numbers added. So here is a case where even numbers are handicapped versus odd numbers and that rarely happens.