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Classification of Interesting Numbers

Lax59.InterestingNumbers · concepts/Lax59/InterestingNumbers.lean · lax-59

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    Natural Language Statement

    Theorem

    Let nn be interesting when there are distinct primes pp and qq such that n=pqn=pq, and when replacing pp by its next prime and qq by its previous prime gives a factorization of n+1n+1. Then nn is interesting exactly when n=14n=14 or n=21n=21.

    The two factors are existentially labelled. This is equivalent to the opposite labelling used in mathlib4 issue #6091 because multiplication is commutative.

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    1import Mathlib
    2
    3/-!
    4---
    5title: Classification of Interesting Numbers
    6type: theorem
    7---
    8Let nn be interesting when there are distinct primes pp and qq such that
    9n=pqn=pq, and when replacing pp by its next prime and qq by its previous prime
    10gives a factorization of n+1n+1. Then nn is interesting exactly when
    11n=14n=14 or n=21n=21.
    12
    13The two factors are existentially labelled. This is equivalent to the
    14opposite labelling used in mathlib4 issue #6091 because multiplication is
    15commutative.
    16-/
    17
    18namespace Lax59.InterestingNumbers
    19
    20/-- The number called `pNext` is the first prime strictly after `p`. -/
    21def NextPrime (p pNext : ℕ) : Prop :=
    22 pNext.Prime ∧ p < pNext ∧ ∀ r, r.Prime → p < r → pNext ≤ r
    23
    24/-- The number called `qPrev` is the last prime strictly before `q`. -/
    25def PrevPrime (qPrev q : ℕ) : Prop :=
    26 qPrev.Prime ∧ qPrev < q ∧ ∀ r, r.Prime → r < q → r ≤ qPrev
    27
    28/-- The neighboring-prime factorization property from issue #6091. -/
    29def Interesting (n : ℕ) : Prop :=
    30 ∃ p q pNext qPrev : ℕ,
    31 p.Prime ∧ q.Prime ∧ p ≠ q ∧
    32 NextPrime p pNext ∧ PrevPrime qPrev q ∧
    33 n = p * q ∧ n + 1 = pNext * qPrev
    34
    35/-- Exactly 14 and 21 satisfy the interesting-number property. -/
    36axiom interesting_iff (n : ℕ) : Interesting n ↔ n = 14 ∨ n = 21
    37
    38end Lax59.InterestingNumbers
    39
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