G-g-generalize

I never meant for it to happen. It violates all intentions of this blog to generalize a theorem. So I’d like to blame Ernie Croot for the suggestion;

Let B be any (unordered) set of integers. Let L_n be the least common multiple of \{1,2,..,n\} and X_n = L_n \displaystyle \sum_{i=1}^n \dfrac{b_i}{i}, where b_i \in B for all i. Then \gcd(X_n, L_n) > 1 necessarily happens infinitely often in exactly 2 cases: there either exists an odd prime p for which b_i \equiv b_j \pmod{p} for all b_i, b_j \in B, or all the b_i are even. And it’s not even necessary to start the sum at i = 1.

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