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I wonder if the triangle hides any secrets related to prime numbers as well.
by zkmon 1y ago
I wonder if the triangle hides any secrets related to prime numbers as well.
- dhosek 1y agoIt’s possible, I suppose. One of my favorite proofs that the sums of each row are powers of two comes from the fact that the numbers in row n+1 are the coefficients of the powers of (a+b)ⁿ, so setting a=b=1 you get 2ⁿ (most discrete math students seeking to prove this end up reaching for induction which is a heavier proof than this).
- nimonian 1y agoI like the argument that every number in the row below is formed by summing two numbers from above. So each number above appears twice below. Hence the sum doubles.
- madcaptenor 1y agoThat requires you to prove the binomial theorem first, though, and won't that need induction?
- dhosek 1y agoDepends on your starting point.
- adornKey 1y agoIndeed it does, e.g: https://nonagon.org/ExLibris/paul-erdos-bertrands-conjecture https://nonagon.org/ExLibris/paul-erdos-bertrands-conjecture