Tough puzzles requiring mathematical skill

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Mathematical Puzzles The Contest Center 59 DeGarmo Hills Road Wappingers Falls, NY 12590

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All of the mathematical puzzles on this webpage are very challenging.
We will post the names of anyone who solves them.
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* indicates I have only a partial solution.
** indicates I do not have a solution.

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** Rubin's Conjecture #1
       Prove or disprove that every integer can be expressed as a sum of 3 distinct powers,   x^a + y^b + z^c   where   x, y, z, a, b, c   are all integers, and   1<a<b<c.
       For a table showing how each of the integers from -100 to +100 can be expressed in this form CLICK HERE. For a table showing how each of the integers from -8000 to +8000 can be expressed in this form CLICK HERE.

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** Rubin's Conjecture #2
       Prove or disprove that all but a finite number of integers can be expressed as a sum of 3 powers,   x^a + y^b + z^c   where   x, y, z, a, b, c   are all positive integers, and   1<a<b<c.

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