FormalConjectures.ErdosProblems.«290»

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import FormalConjecturesUtil

Erdős Problem 290

References:

erdosproblems.com/290

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980), p.34.

[vD24] van Doorn, W., On the non-monotonicity of the denominator of generalized harmonic sums. arXiv:2411.03073 (2024).

namespace Erdos290

The denominator (in lowest terms) of the partial harmonic sum $\sum_{a \leq n \leq b}\frac{1}{n}$.

noncomputable def harmonicDen (a b : ℕ) : ℕ := (harmonicBlock a b).den

Let $a\geq 1$. Must there exist some $b>a$ such that $$\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and } \sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},$$ with $(r_i,s_i)=1$ and $s_2<s_1$? If so, how does this $b(a)$ grow with $a$?

This was resolved in the affirmative by van Doorn [vD24], who proved $b=b(a)$ always exists, and in fact $b(a) \ll a$. Indeed, if $a\in (3^k,3^{k+1}]$ then one can take $b=2\cdot 3^{k+1}-1$. van Doorn also proves that $b(a)>a+(1/2-o(1))\log a$, and considers various generalisations of the original problem.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/Woett/Lean-files/blob/main/ErdosProblem290.lean"]
theorem erdos_290 : answer(True) ↔
    (∀ a : ℕ, 1 ≤ a → ∃ b : ℕ, a < b ∧ harmonicDen a (b + 1) < harmonicDen a b) := by⊢ True ↔ ∀ (a : ℕ), 1 ≤ a → ∃ b, a < b ∧ harmonicDen a (b + 1) < harmonicDen a b
  sorryAll goals completed! 🐙

end Erdos290