Show that every positive rational number can be presented as a ratio of powers' sum in the following way:
(a^{2}+b^{3})/ (c^{5}+d^{7})
where a,b,c,d are positive integers, not necessarily distinct.
Let the rational number be x/y. Let a=x^3*y^2, b=x^5*y^2, c=x*y, and d=x^2*y. Then, (a^2+b^3)/(c^5+d^7)=((x^3*y^2)^2+(x^5*y^2)^3)/((x*y)^5+(x^2*y)^7)=((x^6*y^4)+(x^15*y^6))/((x^5*y^5)+(x^14*y^7))=x/y.

Posted by Math Man
on 20170620 07:36:16 