494.lean`

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

12 declarations, 9 contain sorry, 0 declared as axioms

Erdős Problem 494

References:

erdosproblems.com/494
[SeSt58] Selfridge, J. L. and Straus, E., On the determination of numbers by their sums of a fixed order. Pacific Journal of Math. (1958), 847-856.
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[GFS62] Gordon, B. and Fraenkel, A. S. and Straus, E. G., On the determination of sets by the sets of sums of a certain order. Pacific J. Math. (1962), 187--196.
[FoIz94] Fomin, D. V. and Izhboldin, O. T., Sets of multiple sums. Proc. St. Petersburg Math. Soc. 3 (1994), 244-259.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos494.sumMultiset For a finite set $A \subset \mathbb{C}$ and $k \ge 1$, define $A_k$ as the multiset consisting of all sums of $k$ distinct elements of $A$ line 45 source
Source documentation
For a finite set $A \subset \mathbb{C}$ and $k \ge 1$, define $A_k$ as the multiset consisting of all sums of $k$ distinct elements of $A$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Complex, Nat, Multiset
Proof uses: Multiset.map, Finset.sum, id, Finset.val, Finset.powersetCard
1 instances, coercions and auxiliary constants left out.
```
noncomputable def sumMultiset (A : Finset ℂ) (k : ℕ) : Multiset ℂ :=
…
```
def Erdos494.Erdos494Unique line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset, Complex, Eq, Finset.card, Multiset, Erdos494.sumMultiset
theorem Erdos494.erdos_494.variants.k_eq_2_card_not_pow_two Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 2$ and $|A| \ne 2^l$ for $l \ge 0$ sorry line 57 source
Source documentation
Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 2$ and $|A| \ne 2^l$ for $l \ge 0$. They also gave counterexamples when $k = 2$ and $|A| = 2^l$.

depends on sorry what this means
Statement uses: Nat, Ne, HPow.hPow, OfNat.ofNat, Erdos494.Erdos494Unique
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_494.variants.k_eq_2_card_not_pow_two :
    ∀ card : ℕ, (∀ l : ℕ, card ≠ 2 ^ l) → Erdos494Unique 2 card := by
…
```
Mentions: Erdos494.Erdos494Unique
theorem Erdos494.erdos_494.variants.k_eq_2_card_pow_two sorry line 68 source
depends on sorry what this means
Statement uses: Nat, Exists, Eq, HPow.hPow, OfNat.ofNat, Not, Erdos494.Erdos494Unique
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 Erdos494.erdos_494.variants.k_eq_3_card_gt_6 Selfridge and Straus [SeSt58] also showed that the conjecture is true when 1) $k = 3$ and $|A| > 6$, except possibly for $|A| = 27$ and $|A| = 486$, or 2) $k = 4$ and $|A| > 12$ sorry line 84 source
Source documentation
Selfridge and Straus [SeSt58] also showed that the conjecture is true when
1. $k = 3$ and $|A| > 6$, except possibly for $|A| = 27$ and $|A| = 486$, or
2. $k = 4$ and $|A| > 12$. More generally, they proved that $A$ is determined by $A_k$ (and $|A|$) if $|A|$ is divisible by a prime greater than $k$.
The cases $|A| = 27$ and $|A| = 486$ were left open in [SeSt58]. Fomin and Izhboldin [FoIz94] later found two distinct sets of each of these sizes with the same multiset of $3$-sums, so these exceptions are genuine.
depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Ne, Erdos494.Erdos494Unique
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_494.variants.k_eq_3_card_gt_6 :
    ∀ card > 6, card ≠ 27 → card ≠ 486 → Erdos494Unique 3 card := by
…
```
Mentions: Erdos494.Erdos494Unique
theorem Erdos494.erdos_494.variants.k_eq_4_card_gt_12 Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 4$ and $|A| > 12$ sorry line 93 source
Source documentation
Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 4$ and $|A| > 12$.

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Erdos494.Erdos494Unique
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_494.variants.k_eq_4_card_gt_12 :
    ∀ card > 12, Erdos494Unique 4 card := by
…
```
Mentions: Erdos494.Erdos494Unique
theorem Erdos494.erdos_494.variants.card_divisible_by_prime_gt_k Selfridge and Straus [SeSt58] proved that $A$ is determined by $A_k$ if $|A|$ is divisible by a prime greater than $k$ sorry line 102 source
Source documentation
Selfridge and Straus [SeSt58] proved that $A$ is determined by $A_k$ if $|A|$ is divisible by a prime greater than $k$.

depends on sorry what this means
Statement uses: Nat, Nat.Prime, Membership.mem, Set, Set.Ioo, OfNat.ofNat, Dvd.dvd, Erdos494.Erdos494Unique
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_494.variants.card_divisible_by_prime_gt_k :
    ∀ (k card p : ℕ), p.Prime → k ∈ Set.Ioo 0 p → p ∣ card → Erdos494Unique k card := by
…
```
Mentions: Erdos494.Erdos494Unique
theorem Erdos494.erdos_494.variants.k_eq_card sorry line 112 source
depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Not, Erdos494.Erdos494Unique
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos494.erdos_494.variants.card_eq_2k sorry line 123 source
depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Not, Erdos494.Erdos494Unique, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos494.erdos_494.variants.gordon_fraenkel_straus Gordon, Fraenkel, and Straus [GRS62] proved that the claim is true for all $k > 2$ when $|A|$ is sufficiently large sorry line 132 source
Source documentation
Gordon, Fraenkel, and Straus [GRS62] proved that the claim is true for all $k > 2$ when $|A|$ is sufficiently large.

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Filter.Eventually, Erdos494.Erdos494Unique, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_494.variants.gordon_fraenkel_straus :
    ∀ k > 2, ∀ᶠ card in atTop, Erdos494Unique k card := by
…
```
Mentions: Erdos494.Erdos494Unique
def Erdos494.prodMultiset A version in [Er61] by Erdős is product instead of sum, which is false line 141 source
Source documentation
A version in [Er61] by Erdős is product instead of sum, which is false. Counterexample (by Steinerberger): consider $k = 3$ and let $A = \{1, \zeta_6, \zeta_6^2, \zeta_6^4\}$ and $B = \{1, \zeta_6^2, \zeta_6^3, \zeta_6^4\}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Complex, Nat, Multiset
Proof uses: Multiset.map, Finset.prod, id, Finset.val, Finset.powersetCard
2 instances, coercions and auxiliary constants left out.
```
noncomputable def prodMultiset (A : Finset ℂ) (k : ℕ) : Multiset ℂ :=
…
```
theorem Erdos494.erdos_494.variants.product sorry line 148 source
depends on sorry what this means
Statement uses: Exists, Finset, Complex, And, Eq, Nat, Finset.card, Multiset, Erdos494.prodMultiset, OfNat.ofNat, Ne
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.