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        riddles >> putnam exam (pure math) >> New conjecture
        
(Message started by: aicoped on Sep 11^th, 2011, 5:20pm)

Title: New conjecture
Post by aicoped on Sep 11^th, 2011, 5:20pm

Ok when I was a young lad I was playing around with squares and cubes and whatnot one summer and I discovered and figured out the formula for what i later learned was already called Waring's conjecture.

Briefly for every power there exists a maximal number such that all positive integers can be expressed as the sum of that many or less terms to the given power.

For squares, every positive integer can be expressed as the sum of 4 or less square numbers.

Cubes take 9.

Fourth powers take no more than 19 and so on.

Now here is the interesting thing to me at least. Once numbers start getting relatively large their maximal number for that power goes down(except for squares, which possibly always will always need 4 for every number of the form 8x-1).

With all that being said, here is my conjecture. To my knowledge it is unique to me. if anyone can show me a proof or direct me to someone that conjectured it already, I would be appreciative.

Any sufficiently large number can be expressed as the sum of at most 3 positive integers taken to powers 2 or higher(these powers need not be the same).

for example 127 takes 4 squares to do(11^2+2^2+1^2+1^2), but only 2 numbers if multiple powers are allowed(10^2+3^3).

Title: Re: New conjecture
Post by william wu on Jan 22^nd, 2012, 1:49pm

Usually when I see a conjecture like this, I also want to see a computer printout of some sort --- perhaps generated by a brute force program --- that shows that the conjecture is true for many many numbers before anyone bothers trying to prove that it is true. Do you have such evidence?

Title: Re: New conjecture
Post by SMQ on Feb 13^th, 2012, 2:06pm

I can currently confirm by computer search that the conjecture holds for 1 _{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif} n _{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif} 4http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/times.gif10¹⁰, but I'm not yet confident it holds generally.

Also note that the conjecture can equivalently be stated as "all positive integers of the form 4^s(8t + 7) can be represented as the sum of three perfect powers of non-negative integers" since it is known that all positive integers not of the form 4^s(8t + 7) can be represented as the sum of three squares. (Gauss, after an near-proof by Legendre).

--SMQ