FormalConjectures/Wikipedia/FibonacciPrimes.lean

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Lean4 Domain Explanation Language

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

3 declarations, 2 contain sorry, 0 declared as axioms

Fibonacci Primes

Reference: Wikipedia

Imports (1)
  1. theorem FibonacciPrimes.fib_primes_infinite Documented declaration: fib_primes_infinite sorry line 37 source

    Source documentation

    There are infinitely many Fibonacci primes, i.e., Fibonacci numbers that are prime It is also a barrier to defining a benchmark from this paper: https://arxiv.org/html/2505.13938v1 (see Figure 8).

    depends on sorry what this means

    Statement uses: Set.Infinite, Nat, Set.ofPred, And, Exists, Eq, Nat.fib, Nat.Prime

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    1 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem fib_primes_infinite : {n : ℕ | (∃ m : ℕ, m.fib = n) ∧ n.Prime}.Infinite := by
…
  2. theorem FibonacciPrimes.fib_primes_infinite.variant There are infinitely many indices $i$, such that the $i$-th Fibonacci is prime sorry line 44 source

    Source documentation

    There are infinitely many indices $i$, such that the $i$-th Fibonacci is prime.

    depends on sorry what this means

    Statement uses: Set.Infinite, Nat, Set.ofPred, Nat.Prime, Nat.fib

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat

    1 instances, coercions and auxiliary constants left out.

    Tactic trace: 1 steps

    Open to load the checked before-and-after goal states.

    theorem fib_primes_infinite.variant : {n : ℕ | n.fib.Prime}.Infinite := by
…
  3. theorem FibonacciPrimes.indices_infinite_iff_fib_primes_infinite The two ways of phrasing the conjecture are equivalent line 51 source

    Source documentation

    The two ways of phrasing the conjecture are equivalent.

    standard axioms: Classical.choice, Quot.sound, propext what this means

    Statement uses: Iff, Set.Infinite, Nat, Set.ofPred, Nat.Prime, Nat.fib, And, Exists, Eq

    Proof uses: Eq.mpr, Membership.mem, Set, LT.lt, id, congr, congrArg, Iff.intro, HAdd.hAdd, OfNat.ofNat, Exists.intro, And.intro, exists_apply_eq_apply, Mathlib.Tactic.LinearCombination.lt_of_lt, add_lt_add_of_le_of_lt, LE.le, Nat.le_fib_add_one, Mathlib.Tactic.Ring.le_congr, Nat.rawCast, HMul.hMul and 24 more

    57 instances, coercions and auxiliary constants left out.

    Tactic trace: 18 steps

    Open to load the checked before-and-after goal states.

    theorem indices_infinite_iff_fib_primes_infinite : type_of% fib_primes_infinite.variant ↔
    type_of% fib_primes_infinite := by
…

    Mentions: FibonacciPrimes.fib_primes_infinite.variant, FibonacciPrimes.fib_primes_infinite