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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 Partial solution | Comment 1 of 5
It turns we already had this problem with Dangerous Expression for Rationals but nobody solved it.

After reviewing those comments and mulling it over I had an inspiration.  x^6 is a power of 3 and is one degree higher than x^5.  Similarly, x^8 is a power of 2 and is one degree higher than x^7.

Let a=x^4, b=x^2, c=x, d=x.  Then (a^2+b^3)/(c^5+d^7) = (x^8+x^6)/(x^5+x^7)  = x*(x^7+x^5)/(x^5+x^7) = x.  This works for any nonzero integer x, but does not satisfy the non-integer rationals (x needs to be an integer for a,b,c,d to be integers as stated).

  Posted by Brian Smith on 2017-06-18 10:52:24
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