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Integer equation?



Here's an equation that's easy to analyze with a little insight.
It's not original with me, I ran across it yesterday. (I'll provide attribution after it's solved.)

x3 - y3 = 217

Does this equation have integer solutions (pairs of {x, y} integers)?
If not, prove there are none; if so, find as many as you can.

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x3 - y3 factors into (x - y)(x2 + xy + y2)

Which means if there are any integer solutions, x - y would be an integer and be a factor of 217. There are only 4 factors of 217: 1, 7, 31, 217. We also know that x3 > y3, which means x > y. With that information, we can take a "brute force" method to find all possibilities, knowing that x - y must equal either 1, 7, 31, or 217. It would be something like this:

Assume x - y = 1

  1. Solve for x: x = y + 1
  2. Substitute: 217 = (y + 1 - y)((y + 1)2 + (y + 1)y + y2) = 3y2 + 3y + 1
  3. Simplify: 216 = 3y2 + 3y...72 = y(y + 1)
  4. Solve for y: y=8 or y=-9
  5. Substitute and solve for x to find 2 solutions:
    x=9, y=8
    x=-8, y=-9

Now just repeat for x - y = 7, 31, 217 to find all solutions (note: after going through this, I realized you could probably logically drop 31 and 217 from the list of possible values from the start)...but regardless, the remaining integer solutions are:
x=1, y=-6
x=6, y=-1


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