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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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Solution Solution | Comment 3 of 5 |
Building on the two previous posts I finally found a solution.

Let our rational number be expressed as x/y, with x and y integers.

Then make a=x^3*y^2, b=x^5*y^2, c=x*y, 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*(x^5*y^4+x^14*y^6)) / (y*(x^5*y^4+x^14*y^6))
= x/y

Another similar solution can be found with a=x^4*y^3, b=x^2*y^3, c=x*y^2, d=x*y.

  Posted by Brian Smith on 2017-06-18 20:46:01
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