Show that every positive rational number can be presented as a ratio of powers' sum in the following way:

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(a**^{2}+b^{3})/ (c^{5}+d^{7})

where a,b,c,d are positive integers, not necessarily distinct.

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(In reply to

Partial solution by Brian Smith)

I got integers resolved quickly; integer reciprocals seems to be the next easiest target.

y^4 is a power of 2 and is one degree lower than y^5. Similarly, y^6 is a power of 3 and is one degree lower than y^7.

Let a=y^2, b=y^2, c=y and d=y. Then (a^2+b^3)/(c^5+d^7) = (y^4+y^6)/(y^5+y^7) = (y^4+y^6)/(y*(y^4+y^6)) = 1/y. This works for any nonzero integer y and satisfies the reciprocals of integers.

Is it possible to combine this with the previous parameterization so that all rational x/y work out?