Each positive rational number can be written in the form: (a^2+b^3)/(c^5+d^7)
where a,b,c,d are positive integers.

Assume that instead of the given problem, the question was:

Prove or disprove the following statement;

Each positive rational number can be written in the form: ((2^a)^2+(2^b)^3))/((2^c)^5+(2^d)^7).

Then we can easily find:

4352/17408 =1/4, 576/1152=1/2, 768/1152 = 2/3, 36864/49152=3/4, 2/2=1.

Representing 1/3 might be a bit harder,  but once it is remembered that 2^x+2^x=2^(x+1), while 2^(x+1)+2^(x+2)=3*(2^(x+1), qualifying values such as: ((2^27)^2+(2^18)^3)/((2^11)^(5)+(2^8)^(7)) can be found.

Then we have 2^(x+2)+2^(x) = 5*2^x, 2^(x+3)+2^(x)=9*2^x etc. If we can get this far just with powers of 2, I don't see why it should not be possible to achieve all rational numbers with the entire universe of positive integer a,b,c,d to choose from.

Edited on April 5, 2013, 10:30 am

Posted by broll on 2013-04-05 06:17:39