430.lean`

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

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 430

References:

erdosproblems.com/430
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

The question is equivalent to the first part of Erdős Problem 385 (an observation of Sarosh Adenwalla recorded on both problem pages).

Imports (1)

FormalConjecturesUtil

def Erdos430.terms The terms of the sequence of Erdős Problem 430 for a given $n$: the integers $m$ with $1 < m < n$ all of whose prime factors are $> n - m$ line 48 source
Source documentation
The terms of the sequence of Erdős Problem 430 for a given $n$: the integers $m$ with $1 < m < n$ all of whose prime factors are $> n - m$.
The decreasing sequence $a_1 > a_2 > \cdots$ of the problem lists the elements of this set in decreasing order: $a_1 = n - 1$ is its largest element and $a_k$ is its largest element below $a_{k-1}$. The integer $1$ has no prime factors and is not a term of the sequence, which stops instead, as in the example on erdosproblems.com: for $n = 8$ the sequence is $7, 5$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.filter, Membership.mem, Nat.primeFactors, LT.lt, HSub.hSub, Finset.Ioo, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.
```
def terms (n : ℕ) : Finset ℕ := (Finset.Ioo 1 n).filter fun m => ∀ p ∈ m.primeFactors, n - m < p
…
```
theorem Erdos430.terms_eight For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop line 52 source
Source documentation
For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Finset, Nat, Erdos430.terms, OfNat.ofNat, Insert.insert, Singleton.singleton
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem terms_eight : terms 8 = {5, 7} := by
…
```
Mentions: Erdos430.terms
theorem Erdos430.isGreatest_sub_one The first term of the sequence is $a_1 = n - 1$, the largest element of terms n line 57 source
Source documentation
The first term of the sequence is $a_1 = n - 1$, the largest element of terms n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, IsGreatest, Finset, Erdos430.terms, HSub.hSub
Proof uses: And.intro, Membership.mem, Set, upperBounds, Eq.mpr, And, LT.lt, Nat.primeFactors, id, Eq, Eq.trans, Finset.filter, Finset.Ioo, congrFun', congrArg, Decidable.byContradiction, Not, Nat.Prime.one_lt, Nat.prime_of_mem_primeFactors, Eq.mp
28 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem isGreatest_sub_one {n : ℕ} (hn : 3 ≤ n) : IsGreatest (terms n : Set ℕ) (n - 1) := by
…
```
Mentions: Erdos430.terms
theorem Erdos430.erdos_430 Fix some integer sorry line 74 source
Source documentation
Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$.
Is it true that, for sufficiently large $n$, not all of this sequence can be prime?

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Not, Membership.mem, Finset, Erdos430.terms, Nat.Prime, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_430 : answer(sorry) ↔ ∀ᶠ n in atTop, ¬ ∀ m ∈ terms n, m.Prime := by
…
```
Mentions: Erdos430.terms