Does there exist any positive integral solution of the equation given below?

16(X³+Y³) + 12(X²+Y²) + 3(X+Y) = 2,138,156,388

Yeah, so my calculating skills were broken and I didn't read the problem either.

How I know there are no Positive Integral Solutions:

First of all since x & y would necessarily be greater than zero, x and y are both bounded above by 511

In Excel I put the series 1,2,3,...511 in Column A.

In B1 I put =16*(A1^3)+12*(A1^2)+3*(A1) and filled in the rest of the rows of Column B.

In C1 I put 2,138,156,388-B1 and filled down.

My thinking is that if there exists a solution then there should be one number in Column B which is identical to a number in Column C. To find this, I use the expression:

=FREQUENCY(B:B,C1)-FREQUENCY(B:B,C1-1) in D1 and filled down.

This function returns the number of values in the B column less than or equal to C1 minus the number of values less than C1. That is to say this number will be zero unless there exists a solution. Now when I search Column D for anything other than a zero I find nothing.

Therefore there are no Positive Integral Solutions.

Edited on December 29, 2005, 1:48 am

Edited on December 29, 2005, 1:48 am

Edited on December 29, 2005, 1:50 am

Edited on December 29, 2005, 1:52 am

Posted by Eric on 2005-12-29 01:45:25