Formalize Erdős Problem 1107 #1327
Open
+55
−0
Conversation
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
mo271
reviewed
Dec 4, 2025
Collaborator
mo271
left a comment
mo271
left a comment
There was a problem hiding this comment.
Thanks for the contribution. A few remarks...
Comment on lines
29
to
34
| /-- | ||
| A number $n$ is $r$-powerful if for every prime $p$ which divides $n$, | ||
| we have $p^r$ divides $n$. | ||
| -/ | ||
| def IsRPowerful (r n : ℕ) : Prop := | ||
| ∀ p, p.Prime → p ∣ n → p ^ r ∣ n |
Collaborator
There was a problem hiding this comment.
Suggested change
| /-- | |
| A number $n$ is $r$-powerful if for every prime $p$ which divides $n$, | |
| we have $p^r$ divides $n$. | |
| -/ | |
| def IsRPowerful (r n : ℕ) : Prop := | |
| ∀ p, p.Prime → p ∣ n → p ^ r ∣ n |
Powerful is already definied in
FormalConjectures/ForMathlib/Data/Nat/Full.lean
Comment on lines
41
to
42
| ∀ r ≥ 2, ∃ N, ∀ n ≥ N, | ||
| ∃ s : List ℕ, s.length ≤ r + 1 ∧ (∀ x ∈ s, IsRPowerful r x) ∧ s.sum = n := by |
Collaborator
There was a problem hiding this comment.
Suggested change
| ∀ r ≥ 2, ∃ N, ∀ n ≥ N, | |
| ∃ s : List ℕ, s.length ≤ r + 1 ∧ (∀ x ∈ s, IsRPowerful r x) ∧ s.sum = n := by | |
| ∀ r ≥ 2, ∃ᶠ n : ℕ in Filter.atTop, | |
| ∃ s : List ℕ, s.length ≤ r + 1 ∧ (∀ x ∈ s, r.Full x) ∧ s.sum = n := by |
There is ∃ᶠ and Full
| ∀ r ≥ 2, ∃ N, ∀ n ≥ N, | ||
| ∃ s : List ℕ, s.length ≤ r + 1 ∧ (∀ x ∈ s, IsRPowerful r x) ∧ s.sum = n := by | ||
| sorry | ||
|
|
Collaborator
There was a problem hiding this comment.
Suggested change
| /-- | |
| Every large integer the sum of at most $2 + 1$ many $2$-powerful numbers, | |
| as proved by Heath-Brown [He88] | |
| [He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988) | |
| -/ | |
| @[category research solved, AMS 11] | |
| theorem erdos_1107.variants.two : ∃ᶠ n : ℕ in Filter.atTop, | |
| ∃ s : List ℕ, s.length ≤ 2 + 1 ∧ (∀ x ∈ s, Powerful x) ∧ s.sum = n := by | |
| sorry |
Let's add this known result as well. One could also define the Property ∃ᶠ n : ℕ in Filter.atTop, ∃ s : List ℕ, s.length ≤ r + 1 ∧ (∀ x ∈ s, r.Full x) ∧ s.sum = n separately and re-use it for both theorems. That would be nicer, but whatever works, I suppose.
Author
|
Thanks for the feedback! I'll update the docstring to include the Heath-Brown (1988) result and add the erdos_1107.variants.two theorem as requested. I'll also add the solved category tag. Working on the updates now! |
…wn variant Address code review feedback from @mo271: - Replaced local `IsRPowerful` definition with `Nat.Full` imported from `ForMathlib`. - Refactored the summation property into a helper `SumOfRPowerful` to avoid code duplication. - Switched to filter notation (`∀ᶠ n in atTop`) to correctly capture the "every large integer" condition. - Added `erdos_1107.variants.two` to formally state the solved case for r=2 (Heath-Brown, 1988). - Updated docstrings and tags (added `solved` to the variant).
…mal-conjectures into formalize-erdos-1107
mo271
reviewed
Dec 4, 2025
| sorry | ||
|
|
||
| /-- | ||
| Heath-Brown [He88] proved this for $r=2$. |
Collaborator
There was a problem hiding this comment.
Suggested change
| Heath-Brown [He88] proved this for $r=2$. | |
| Heath-Brown [He88] proved every large integer the sum of at most three | |
| $2$-powerful numbers. |
Let's repeat what "this" is here
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
2 participants
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
Closes #1304
Summary
This PR adds the formalization for Erdős Problem 1107 involving r-powerful numbers.
Changes
FormalConjectures/Erdos1107.leanIsRPowerfulVerification