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 Estimate a percentage (Posted on 2016-08-22)
What percentage of the numbers from one to a million can be represented as the sum of a square and a non-negative cube?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 re: A simple upperbound | Comment 4 of 6 |
(In reply to A simple upperbound by Jer)

Think of sqrt(1000000-C^3) as a function S(C).  Then the curve defines the maximum value a square can have and still make the sum S^2+C^3 not exceed 1000000.

The rectangle with vertecies (0,0), (100,0), (0,1000), (100,1000) then includes all possible pairs (C,S) whose individual powers are under 1000000 and then the curve S(C) includes just those pairs whose sum of powers is under 1000000.

Then numerically integrating S(C) from 0 to 100 yields 84131 when rounded to an integer.  This is close to the upper limit of 84545 that Jer calculated discreetly.
 Posted by Brian Smith on 2017-07-16 23:38:06

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