477.lean`

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

5 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 477

References:

erdosproblems.com/477
Sek59 Milan Sekanina, Замечания к фактoризации беcкoнечнoй цикличеcкoй группы, Czechoslovak Mathematical Journal, Vol. 9 (1959), No. 4, 485–495
The resolution is recorded at erdosproblems.com/477, with a proof exposition by T. F. Bloom of a construction found independently by several provers.

Imports (1)

FormalConjecturesUtil

theorem Erdos477.erdos_477 sorry line 50 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Polynomial, Int, And, LE.le, WithBot, Nat, OfNat.ofNat, Polynomial.degree, Set, ExistsUnique, Prod, Membership.mem, SProd.sprod, Set.range, Polynomial.eval, Eq, HAdd.hAdd and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos477.erdos_477.variants.S_sq There is no such $A$ for the polynomial $f(x) = X^2$ sorry line 61 source
Source documentation
There is no such $A$ for the polynomial $f(x) = X^2$.
This is shown in [Sek59].

depends on sorry what this means
Statement uses: Set, Int, Exists, Not, ExistsUnique, Prod, And, Membership.mem, SProd.sprod, Set.range, Polynomial.eval, HPow.hPow, Polynomial, Nat, Polynomial.X, OfNat.ofNat, Eq, HAdd.hAdd, Prod.fst, Prod.snd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_477.variants.S_sq :
    letI f := X ^ 2
…
```
theorem Erdos477.erdos_477.variants.degree_two_dvd_condition_b_ne_zero There is no such $A$ for any polynomial $f(x) = aX^2 + bX + c$, if $a | b$ with $a \ne 0$ and $b \ne 0$ sorry line 72 source
Source documentation
There is no such $A$ for any polynomial $f(x) = aX^2 + bX + c$, if $a | b$ with $a \ne 0$ and $b \ne 0$. This was found be AlphaProof for the specific instance $X^2 - X + 1$ and then generalised.

depends on sorry what this means
Statement uses: Int, Ne, OfNat.ofNat, Dvd.dvd, Polynomial, HAdd.hAdd, HSMul.hSMul, HPow.hPow, Nat, Polynomial.X, RingHom, Polynomial.C, Set, Exists, Not, ExistsUnique, Prod, And, Membership.mem, SProd.sprod and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_477.variants.degree_two_dvd_condition_b_ne_zero {a b c : ℤ} (ha : a ≠ 0) (hb : b ≠ 0)
    (hab : a ∣ b) :
    let f := a • X ^ 2 + b • X + C c
…
```
Mentions: c
theorem Erdos477.erdos_477.variants.X_pow_three Probably there is no such $A$ for the polynomial $X^3$ sorry line 82 source
Source documentation
Probably there is no such $A$ for the polynomial $X^3$.

depends on sorry what this means
Statement uses: Set, Int, Exists, Not, ExistsUnique, Prod, And, Membership.mem, SProd.sprod, Set.range, Polynomial.eval, HPow.hPow, Polynomial, Nat, Polynomial.X, OfNat.ofNat, Eq, HAdd.hAdd, Prod.fst, Prod.snd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_477.variants.X_pow_three :
    letI f := X ^ 3
…
```
theorem Erdos477.erdos_477.variants.monomial sorry line 94 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Set, Int, Exists, Not, ExistsUnique, Prod, And, Membership.mem, SProd.sprod, Set.range, Polynomial.eval, HPow.hPow, Polynomial, Polynomial.X, Eq and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

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