290.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

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).

Imports (1)

FormalConjecturesUtil

def Erdos290.harmonicDen The denominator (in lowest terms) of the partial harmonic sum $\sum_{a \leq n \leq b}\frac{1}{n}$ line 37 source
Source documentation
The denominator (in lowest terms) of the partial harmonic sum $\sum_{a \leq n \leq b}\frac{1}{n}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Rat.den, harmonicBlock
```
noncomputable def harmonicDen (a b : ℕ) : ℕ := (harmonicBlock a b).den
…
```
Mentions: harmonicBlock
theorem Erdos290.erdos_290 Let $a\geq 1$ sorry line 51 source
Source documentation
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.

L4DEL explanation
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 $s_2a+(1/2-o(1))\log a$, and considers various generalisations of the original problem.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, And, LT.lt, Erdos290.harmonicDen, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_290 : answer(True) ↔
    (∀ a : ℕ, 1 ≤ a → ∃ b : ℕ, a < b ∧ harmonicDen a (b + 1) < harmonicDen a b) := by
…
```
Mentions: Erdos290.harmonicDen