[seqfan] Re: A062682

David Wilson davidwwilson at comcast.net
Sun Nov 18 23:15:31 CET 2012


For cubic or lower degree polynomials, p(x, y) = c in general seems to 
have an unbounded number of solutions, e.g. there are c for which a^3 + 
b^3 = c has an arbitrarily large number of solutions, vis A011541.

For quartic or higher polynomials, this does not appear to be the case. 
For example, a^4 + b^4 = c seems to have at most two distinct solutions 
in a, b for any c, vis A018768.  At the time I computed A018768, I 
computed it way  beyond the published values, but found no values with 
more than two representations. To my knowledge, no one has ever proved 
that there cannot be three representations, but no one has ever found an 
example.

sum(a..b, i^3) = p(b) - p(a-1) where p(n) = (n*(n+1)/2)^2, a quartic.  
Thus I might expect sum(a..b, i^3) = c to have a bounded number of 
distinct solutions.

On 11/18/2012 11:02 AM, Uwe Lauth wrote:
> I wrote a C program to find more missing numbers
> n = sum(i=a..b i^3) = sum(i=a'..b' i^3) where b'>b
> Up to b=2450, nothing new was found.
>
> All numbers I found had only two different ways of this sum.
> What is the first number that has three sums?
>
> Uwe
>
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