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UNSOLVED DIOPHANTINE EQUATIONS
2. The Beal Conjecture. If A^{x} +B^{y} = C^{z} , where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common factor. $75,000 prize.
3. Is there a 3x3 magic square, with 9 distinct entries, so that every entry is a square? Martin Gardner offers $100 for a resolution of this question.
4. Catalan Problem: a^b - c^d = 1 Is there a solution besides a = d = 3, b = c = 2 (3^2 - 2^3 = 1)?
5. 1^n + 2^n +... + k^n = (k+1)^n Solution for n>1 ?
6. (x+y+z)^3 = xyz Non-trivial solutions? (Sierpinski)
7. Euler's Brick. Is there a Integral Rect. Parallelepiped? a^2 + b^2 = c^2 a^2 + d^2 = e^2 b^2 + d^2 = f^2 b^2 + e^2 = g^2
8. Characterize those d for which x^2 - dy^2 = -1 has a solution.
Mathworld has an extensive page on Diophantine
Equations.
http://www.math.niu.edu/~rusin/papers/known-math/index/11DXX.html
http://sisenis.com.latnet.lv/~podnieks/gt4.html
http://www.mathsoft.com/asolve/fermat/fermat.html