$\displaystyle a,b,c \in \mathbb{N}^+, \quad d = 2a^2 = 3b^3+2 = 5c^5+3$

Find the smallest $\displaystyle d$ above

What if $\displaystyle d = 2a^2+1 = 3b^3+2 = 5c^5+3$ ?

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- May 4th 2010, 08:33 AMelimSolve a Integer equation system
$\displaystyle a,b,c \in \mathbb{N}^+, \quad d = 2a^2 = 3b^3+2 = 5c^5+3$

Find the smallest $\displaystyle d$ above

What if $\displaystyle d = 2a^2+1 = 3b^3+2 = 5c^5+3$ ? - May 4th 2010, 12:49 PMOpalg
- May 4th 2010, 04:09 PMelim
Thanks Opalg! Let's forget that half of the problem

What about the smallest $\displaystyle d=2a^2=3b^3=5c^5+3$? - May 4th 2010, 08:06 PMelim
By Chinese Remainder Theorem,

$\displaystyle d = 38+30k$ for some $\displaystyle k$ - May 7th 2010, 04:52 PMelim
Integer solution for $\displaystyle d=2a^2=3b^3+2=5c^5+3$

become very tough. I heard that $\displaystyle d < 10^{50}$ has no solutions - May 8th 2010, 08:11 AMhollywood
So $\displaystyle \frac{d}{2}$ is a square, $\displaystyle \frac{d-2}{3}$ is a cube, and $\displaystyle \frac{d-3}{5}$ is a fifth power.

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

- Hollywood