New answers tagged number-theory
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I think I've got it :
Got to say the hint helps a lot
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We have
$$7^{2n} = \frac{7^{2n}+1}{2} + \frac{7^{2n}-1}{2}
= (\frac{7^{2n}+1}{2} - \frac{7^{2n}-1}{2}) \cdot (\frac{7^{2n}+1}{2} + \frac{7^{2n}-1}{2}) = (\frac{7^{2n}+1}{2})^2 - (\frac{7^{2n}-1}{2})^2$$
$n=1: 7^2 = 25^2-24^2$
$n=2: 7^4 = 49^2 = 1201^2 - 1200^2$
$n=3: 7^6 = 343^2 = 58825^2 - 58824^2$
$n=4: 7^8 = 2401^2 = 2882401^2 - 2882400^2$
$n=5: 7^{10}...
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There is no unique way to label the vertices. So does the question mean “minimum” for a specific consistent cube? Or “minimum” over all consistent cubes? I choose the former because it’s more interesting, and it subsumes the other question
If the vertices 1&2 are diagonally opposite one another, then the labels 1...5 are fixed, up to rotation and ...
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