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Math Help - Solve a Integer equation system

  1. #1
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    Solve a Integer equation system

    a,b,c \in \mathbb{N}^+, \quad d = 2a^2 = 3b^3+2 = 5c^5+3
    Find the smallest d above
    What if d = 2a^2+1 = 3b^3+2 = 5c^5+3 ?
    Last edited by elim; May 4th 2010 at 08:02 PM.
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  2. #2
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    Quote Originally Posted by elim View Post
    a,b,c \in \mathbb{N}^+, \quad d = 2a^2+1 = 3b^3+2 = 5c^5+3
    Find the smallest d above
    If 2a^2+1 = 3b^2+2 then 2a^2\equiv 1\!\!\!\pmod3, which is impossible. So no solutions there. But the remaining equations  d = 2a^2+1 = 5c^5+3 have at least one solution d = 163 (with a = 9 and c = 2).
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  3. #3
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    Thanks Opalg! Let's forget that half of the problem
    What about the smallest d=2a^2=3b^3=5c^5+3?
    Last edited by elim; May 4th 2010 at 08:04 PM.
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    By Chinese Remainder Theorem,
    d = 38+30k for some k
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  5. #5
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    Integer solution for d=2a^2=3b^3+2=5c^5+3
    become very tough. I heard that d < 10^{50} has no solutions
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    So \frac{d}{2} is a square, \frac{d-2}{3} is a cube, and \frac{d-3}{5} is a fifth power.

    If you look at these conditions modulo some small numbers, you might find something.

    - Hollywood
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