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Rational number's conversion (Posted on 2017-06-18) Difficulty: 4 of 5
Show that every positive rational number can be presented as a ratio of powers' sum in the following way:

(a2+b3)/ (c5+d7)

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

No Solution Yet Submitted by Ady TZIDON    
Rating: 5.0000 (1 votes)

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Some Thoughts re: Partial solution | Comment 2 of 5 |
(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?

  Posted by Brian Smith on 2017-06-18 12:31:58
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