Find all positive integers $n$ and $k$, such that
$$ 1+ {n\choose 2} + {n\choose 4} = 2^k$$
The question was proposed in this YouTube video: https://www.youtube.com/watch?v=YtkIWDE36qU
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1$(n, k) = (2, 1), (3, 2), (4, 3), (5, 4)$. – Circuit Sage Nov 05 '24 at 05:13
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1Another solution is $(n,k) = (10,8)$. I checked up to $n=10^7$ and found no other solutions. – Salmonella mayonnaise Nov 05 '24 at 16:13
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The LHS is related to Moser's circle problem, but the Diophantine equation may be difficult. See also the references at the OEIS-entry. – Dietrich Burde Nov 05 '24 at 20:24