Residues of Lucas numbers

DefinitionClassification of moduli for which the Lucas numbers attain every residue#✓fully proved

The proposition that for every integer \(m > 1\), with \(L_n\) the Lucas numbers, every residue modulo \(m\) is attained (for every \(r\) with \(0 \le r < m\) there is \(n \ge 0\) with \(L_n \equiv r \pmod m\)) if and only if \(m \in \{2, 4, 6, 7, 14\}\) or \(m = 3^k\) for some integer \(k \ge 1\).

def Statement : Prop :=
  ∀ (m : ℕ), 1 < m →
    ((∀ r : ℕ, r < m → ∃ n : ℕ, MetaMathlibExt.lucasNumber n % m = r) ↔
      (m = 2 ∨ m = 4 ∨ m = 6 ∨ m = 7 ∨ m = 14 ∨
        ∃ k : ℕ, 1 ≤ k ∧ m = 3 ^ k))

Sources

• OPG-37402, Lucas Numbers Modulo m [OpenGarden].

Uses 1 · Used by 1

Statement usesLucas numbers

Used byAvila–Chen theorem on Lucas numbers modulo \(m\)

TheoremAvila–Chen theorem on Lucas numbers modulo \(m\)#●ready to prove

Let \(L_n\) denote the Lucas numbers. For every integer \(m > 1\), every residue modulo \(m\) is attained by the Lucas numbers (for every \(r\) with \(0 \le r < m\) there is \(n \ge 0\) with \(L_n \equiv r \pmod m\)) if and only if \(m \in \{2, 4, 6, 7, 14\}\) or \(m = 3^k\) for some integer \(k \ge 1\).

theorem_wanted Statement_holds : Statement

Sources

• Brandon Avila and Yongyi Chen, On Moduli For Which the Lucas Numbers Contain a Complete Residue System, Fibonacci Quarterly 51(2) (2013), 151–152, https://doi.org/10.1080/00150517.2013.12427958.

Uses 1

Statement usesClassification of moduli for which the Lucas numbers attain every residue

1 fully proved1 ready to prove

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