276.lean`

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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 276

References: erdosproblems.com/276

Imports (1)

FormalConjecturesUtil

def Erdos276.IsLucasSequence We define a Lucas sequence to be a Fibonacci sequence with arbitrary starting points L 0 and L 1 line 41 source
Source documentation
We define a Lucas sequence to be a Fibonacci sequence with arbitrary starting points L 0 and L 1.
TODO: There seems to be multiple definitions in the literature, some of which also allow coefficients in the recurrence relation. For now this simple definition has been chosen as it agrees best with the Erdős problem in this same file. However before moving this into ForMathlib one should make a conscious decision about which definition to choose.

no axioms what this means
Statement uses: Nat
Proof uses: Eq, HAdd.hAdd, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
```
def IsLucasSequence (L : ℕ → ℕ) : Prop := ∀ n, L (n + 2) = L (n + 1) + L n
…
```
theorem Erdos276.erdos_276 Is there an infinite Lucas sequence sorry line 49 source
Source documentation
Is there an infinite Lucas sequence $a_0, a_1, \ldots$ where $a_{n+2} = a_{n+1} + a_n$ for $n \ge 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, Erdos276.IsLucasSequence, Nat.Composite, GT.gt, OfNat.ofNat, Eq, Nat.gcd
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_276 : answer(sorry) ↔
    ∃ (a : ℕ → ℕ),
    IsLucasSequence a ∧ (∀ k, (a k).Composite) ∧ (∀ n > 1, ∃ k, Nat.gcd n (a k) = 1) := by
…
```
Mentions: Erdos276.IsLucasSequence