Absolute Lucas pseudoprimes

TheoremJacobi symbols of a discriminant \(D\) agree at odd primes congruent modulo \(D\)#●ready to prove

Let \(p\) and \(p'\) be odd primes and let \(D\) be an integer with \(D \equiv 0\) or \(1 \pmod 4\) that is coprime to \(p\). If \(p' = p + D x\) for some integer \(x\), then the Jacobi symbols agree: \(\left(\frac{D}{p'}\right) = \left(\frac{D}{p}\right)\).

For an odd prime \(p\) the Jacobi symbol \(\left(\frac{D}{p}\right)\) is the Legendre symbol and agrees with the Kronecker symbol.

public theorem_wanted jacobiSym_prime_add_mul_eq {p p' : ℕ} (hp : p.Prime) (hp2 : p ≠ 2)
    (hp' : p'.Prime) (hp'2 : p' ≠ 2) {D : ℤ} (hD : D % 4 = 0 ∨ D % 4 = 1)
    (hDp : IsCoprime D p) {x : ℤ} (hx : (p' : ℤ) = p + D * x) :
    jacobiSym D p' = jacobiSym D p

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Lemma 1.

TheoremBallot's Lemma 2: three divisibilities as one congruence#●ready to prove

Let \(\varepsilon_1, \varepsilon_2, \varepsilon_3 \in \{1, -1\}\), let \(c_1, c_2, c_3\) be pairwise coprime positive integers, let \(g \ne 0\) be an integer, and put \(n = (c_1 g + \varepsilon_1)(c_2 g + \varepsilon_2)(c_3 g + \varepsilon_3)\) and \(\varepsilon_n = \varepsilon_1 \varepsilon_2 \varepsilon_3\). Then \(c_i g \mid n - \varepsilon_n\) for \(i = 1, 2, 3\) if and only if

\[c_1 c_2 c_3 \mid g (\varepsilon_3 c_1 c_2 + \varepsilon_2 c_1 c_3 + \varepsilon_1 c_2 c_3) + \varepsilon_n (\varepsilon_1 c_1 + \varepsilon_2 c_2 + \varepsilon_3 c_3).\]

public theorem_wanted mul_dvd_prod_sub_sign_iff {ε₁ ε₂ ε₃ c₁ c₂ c₃ g : ℤ}
    (hε₁ : ε₁ = 1 ∨ ε₁ = -1) (hε₂ : ε₂ = 1 ∨ ε₂ = -1) (hε₃ : ε₃ = 1 ∨ ε₃ = -1)
    (hc₁ : 0 < c₁) (hc₂ : 0 < c₂) (hc₃ : 0 < c₃)
    (h₁₂ : IsCoprime c₁ c₂) (h₁₃ : IsCoprime c₁ c₃) (h₂₃ : IsCoprime c₂ c₃) (hg : g ≠ 0) :
    (c₁ * g ∣ (c₁ * g + ε₁) * (c₂ * g + ε₂) * (c₃ * g + ε₃) - ε₁ * ε₂ * ε₃ ∧
      c₂ * g ∣ (c₁ * g + ε₁) * (c₂ * g + ε₂) * (c₃ * g + ε₃) - ε₁ * ε₂ * ε₃ ∧
      c₃ * g ∣ (c₁ * g + ε₁) * (c₂ * g + ε₂) * (c₃ * g + ε₃) - ε₁ * ε₂ * ε₃) ↔
    c₁ * c₂ * c₃ ∣
      g * (ε₃ * c₁ * c₂ + ε₂ * c₁ * c₃ + ε₁ * c₂ * c₃) +
        ε₁ * ε₂ * ε₃ * (ε₁ * c₁ + ε₂ * c₂ + ε₃ * c₃)

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Lemma 2.

Definition\(D\)-Carmichael number#✓fully proved

Let \(D\) be an integer. A natural number \(n\) is a \(D\)-Carmichael number when \(D \equiv 0\) or \(1 \pmod 4\), \(n > 1\) is odd and composite, \(D\) and \(n\) are coprime, and for all integers \(P\) and \(Q \ne 0\) with \(P\) and \(Q\) coprime, \(P^2 - 4Q = D\) and \(Q\) coprime to \(n\), the number \(n\) divides the term \(U_{n - \varepsilon_n}(P, Q)\) of the Lucas sequence \(U(P, Q)\), where \(\varepsilon_n = \left(\frac{D}{n}\right)\) is the Jacobi symbol.

These are Williams' absolute Lucas \(D\)-pseudoprimes. The condition on \(D\) is the paper's standing assumption; for other \(D\) no Lucas sequence has discriminant \(D\), so the last clause would be vacuous.

def IsDCarmichael (D : ℤ) (n : ℕ) : Prop :=
  (D % 4 = 0 ∨ D % 4 = 1) ∧ Odd n ∧ 1 < n ∧ ¬ n.Prime ∧ IsCoprime D n ∧
    ∀ params : LucasSequenceParams, params.IsCoprime → params.p ^ 2 - 4 * params.q = D →
      IsCoprime params.q n → (n : ℤ) ∣ lucasU params (n - jacobiSym D n).toNat

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Section 1 (definition of absolute Lucas D-pseudoprimes, after [17]).

Uses 3 · Used by 17

Statement usesParameters of a Lucas sequence · Coprime Lucas sequence parameters · Lucas sequence \(U(P, Q)\)

Used byPrimary \(D\)-Carmichael numbers are \(D\)-Carmichael · Williams' Korselt criterion for \(D\)-Carmichael numbers · Williams' characterization of two-prime \(D\)-Carmichael numbers · Bi-Carmichael numbers for signatures of opposite signs · Primary \(D\)-Carmichael number · Secondary \(D\)-Carmichael number · Carmichael numbers of a signature · Universal \(D\)-forms · Chernick-type universal forms for three prime factors · Three-prime-factor \(D\)-Carmichael numbers have a single Jacobi sign · \(9\) is not a \(5\)-Carmichael number · \(4\) is not a signature Carmichael number · \(x(x+1)\) is never a universal form · \(D\)-Carmichael numbers are not prime · Divisibility conditions for \(D\)-Carmichael numbers \(3qr\) · The \(D\)-Carmichael numbers of the form \(3qr\) · Two-prime-factor \(D\)-Carmichael numbers

Theorem\(D\)-Carmichael numbers are not prime#✓fully proved

For every integer \(D\) and every natural number \(n\), if \(n\) is a \(D\)-Carmichael number, then \(n\) is not prime.

theorem IsDCarmichael.not_prime {D : ℤ} {n : ℕ} (h : IsDCarmichael D n) : ¬ n.Prime

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Section 1.

Uses 4

Statement uses\(D\)-Carmichael number

Proof usesParameters of a Lucas sequence · Coprime Lucas sequence parameters · Lucas sequence \(U(P, Q)\)

Theorem\(9\) is not a \(5\)-Carmichael number#✓fully proved

The number \(9\) is not a \(D\)-Carmichael number for \(D = 5\). Indeed, the Jacobi symbol \((5 \mid 9)\) equals \(1\), and the Fibonacci sequence \(U(1, -1)\) has \(F_8 = 21\), which \(9\) does not divide.

theorem not_isDCarmichael_five_nine : ¬ IsDCarmichael 5 9

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Section 1.

Uses 6

Statement uses\(D\)-Carmichael number

Proof usesParameters of a Lucas sequence · Coprime Lucas sequence parameters · Fibonacci parameters for Lucas sequences · Lucas sequence \(U(P, Q)\) · The Lucas sequence \(U(1, -1)\) is the Fibonacci sequence

DefinitionExtended digit sum#✓fully proved

Let \(D\) be an integer and let \(p \ge 2\) and \(n\) be natural numbers, with base-\(p\) expansion \(n = \sum_i c_i p^i\), \(0 \le c_i < p\). With \(\varepsilon_p = \left(\frac{D}{p}\right)\) the Jacobi symbol, the extended digit sum of \(n\) is \(s_p(n) = \sum_i c_i \varepsilon_p^i\).

For \(p \le 1\) the definition gives \(s_p(n) = n\).

def lucasDigitSum (D : ℤ) (p n : ℕ) : ℤ :=
  ((Nat.digits p n).mapIdx fun i c => (c : ℤ) * jacobiSym D p ^ i).sum

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, equation (6).

Used by 4

Used byPrimary \(D\)-Carmichael numbers are \(D\)-Carmichael · Primary \(D\)-Carmichael number · Secondary \(D\)-Carmichael number · The extended digit sum of \(6601\) in base \(7\) for \(D = 5\)

TheoremThe extended digit sum of \(6601\) in base \(7\) for \(D = 5\)#✓fully proved

The base-\(7\) expansion is \(6601 = 5 \cdot 7 + 7^2 + 5 \cdot 7^3 + 2 \cdot 7^4\) and the Jacobi symbol is \(\left(\frac{5}{7}\right) = -1\), so the extended digit sum with \(D = 5\) is \(s_7(6601) = -5 + 1 - 5 + 2 = -7\).

theorem lucasDigitSum_five_seven_6601 : lucasDigitSum 5 7 6601 = -7

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, equation (6) and the example m = 6601.

Uses 1

Statement usesExtended digit sum

DefinitionPrimary \(D\)-Carmichael number#✓fully proved

Let \(D\) be an integer. A natural number \(n\) is a primary \(D\)-Carmichael number when \(n\) is a \(D\)-Carmichael number and \(|s_p(n) - \varepsilon_n| \le p - \varepsilon_p\) for every prime \(p \mid n\), where \(\varepsilon_m = \left(\frac{D}{m}\right)\) is the Jacobi symbol and \(s_p(n) = \sum_i c_i \varepsilon_p^i\) is the extended digit sum of the base-\(p\) expansion \(n = \sum_i c_i p^i\).

In the paper's terms, \(n\) has degree at most one: the number \(d_p\) with \(s_p(n) = \varepsilon_n + d_p (p - \varepsilon_p)\) satisfies \(|d_p| \le 1\) for every prime \(p \mid n\).

def IsPrimaryDCarmichael (D : ℤ) (n : ℕ) : Prop :=
  IsDCarmichael D n ∧ ∀ p ∈ n.primeFactors,
    |lucasDigitSum D p n - jacobiSym D n| ≤ p - jacobiSym D p

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Definition 3.

Uses 2 · Used by 7

Statement uses\(D\)-Carmichael number · Extended digit sum

Used byPrime \(k\)-tuples give primary \(5\)-Carmichael numbers of every three-term signature · Schinzel's Hypothesis H gives numbers that are primary \(5\)- and \((-11)\)-Carmichael · Primary \(D\)-Carmichael numbers are \(D\)-Carmichael · Primary and secondary \(5\)-Carmichael numbers from prime quadruples · Primary and secondary \(13\)-Carmichael numbers from prime quadruples · Two-prime-factor \(D\)-Carmichael numbers · Universal \(5\)- and \(13\)-forms from prime quadruples

DefinitionSecondary \(D\)-Carmichael number#✓fully proved

Let \(D\) be an integer. A natural number \(n\) is a secondary \(D\)-Carmichael number when \(n\) is a \(D\)-Carmichael number, \(|s_p(n) - \varepsilon_n| \le 2 (p - \varepsilon_p)\) for every prime \(p \mid n\), and \(|s_p(n) - \varepsilon_n| > p - \varepsilon_p\) for some prime \(p \mid n\), where \(\varepsilon_m = \left(\frac{D}{m}\right)\) is the Jacobi symbol and \(s_p(n) = \sum_i c_i \varepsilon_p^i\) is the extended digit sum of the base-\(p\) expansion \(n = \sum_i c_i p^i\).

In the paper's terms, \(n\) has degree exactly two: the number \(d_p\) with \(s_p(n) = \varepsilon_n + d_p (p - \varepsilon_p)\) satisfies \(|d_p| \le 2\) for every prime \(p \mid n\) and \(|d_p| > 1\) for some.

def IsSecondaryDCarmichael (D : ℤ) (n : ℕ) : Prop :=
  IsDCarmichael D n ∧
    (∀ p ∈ n.primeFactors, |lucasDigitSum D p n - jacobiSym D n| ≤ 2 * (p - jacobiSym D p)) ∧
    ∃ p ∈ n.primeFactors, p - jacobiSym D p < |lucasDigitSum D p n - jacobiSym D n|

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Definition 3.

Uses 2 · Used by 3

Statement uses\(D\)-Carmichael number · Extended digit sum

Used byPrimary and secondary \(5\)-Carmichael numbers from prime quadruples · Primary and secondary \(13\)-Carmichael numbers from prime quadruples · Universal \(5\)- and \(13\)-forms from prime quadruples

TheoremPrimary \(D\)-Carmichael numbers are \(D\)-Carmichael#✓fully proved

For every integer \(D\) and natural number \(n\), if \(n\) is a primary \(D\)-Carmichael number, then \(n\) is a \(D\)-Carmichael number.

theorem IsPrimaryDCarmichael.isDCarmichael {D : ℤ} {n : ℕ} (h : IsPrimaryDCarmichael D n) :
    IsDCarmichael D n

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Definition 3.

Uses 3

Statement uses\(D\)-Carmichael number · Primary \(D\)-Carmichael number

Proof usesExtended digit sum

DefinitionCarmichael numbers of a signature#✓fully proved

Let \(\sigma = (\varepsilon_1, \dots, \varepsilon_r)\) be a list of integers. A natural number \(n\) is \(\sigma\)-Carmichael when every \(\varepsilon_i\) is \(1\) or \(-1\), \(r \ge 2\), \(n\) is odd and squarefree with exactly \(r\) prime factors \(p_1 < \dots < p_r\), and \(n\) is \(D\)-Carmichael for every integer \(D \equiv 0\) or \(1 \pmod 4\) whose Jacobi symbols satisfy \(\left(\frac{D}{p_i}\right) = \varepsilon_i\) for all \(i\).

def IsSignatureCarmichael (σ : List ℤ) (n : ℕ) : Prop :=
  Squarefree n ∧ Odd n ∧ 2 ≤ σ.length ∧ n.primeFactorsList.length = σ.length ∧
    (∀ e ∈ σ, e = 1 ∨ e = -1) ∧
    ∀ D : ℤ, (D % 4 = 0 ∨ D % 4 = 1) →
      n.primeFactorsList.map (fun p => jacobiSym D p) = σ → IsDCarmichael D n

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Definition 4.

Uses 1 · Used by 2

Statement uses\(D\)-Carmichael number

Used byKorselt criterion for \(\sigma\)-Carmichael numbers · \(4\) is not a signature Carmichael number

Theorem\(4\) is not a signature Carmichael number#✓fully proved

For every finite sequence \(\sigma\) of integers, \(4\) is not a \(\sigma\)-Carmichael number, because \(4\) is not squarefree.

theorem not_isSignatureCarmichael_four (σ : List ℤ) : ¬ IsSignatureCarmichael σ 4

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Definition 4.

Uses 2

Statement usesCarmichael numbers of a signature

Proof uses\(D\)-Carmichael number

DefinitionUniversal \(D\)-forms#✓fully proved

Let \(D\) be an integer, \(\sigma\) a list of integers, and \(f(x) = (a_1 x + b_1) \cdots (a_k x + b_k)\) a product of linear factors with integer coefficients. Then \(f\) is a universal \(D\)-form with respect to \(\sigma\) when: every factor has \(a_i \ne 0\) and \(\gcd(a_i, b_i) = 1\); for every prime \(\ell\) there is an integer \(x\) with \(\ell \nmid f(x)\); and for every integer \(x\) at which every factor \(a_i x + b_i\) is a positive prime, the product \(f(x)\) is \(D\)-Carmichael and the Jacobi symbols \(\left(\frac{D}{p}\right)\), for \(p\) running through the prime factors of \(f(x)\) in increasing order and counted with multiplicity, form the list \(\sigma\).

Here \(x\) ranges over all integers rather than \(x \ge 0\), since forms such as \(p + D x\) are used with \(D < 0\).

def IsUniversalLinearDForm (D : ℤ) (σ : List ℤ) (l : List (ℤ × ℤ)) : Prop :=
  (∀ ab ∈ l, ab.1 ≠ 0 ∧ Int.gcd ab.1 ab.2 = 1) ∧
    (∀ ℓ : ℕ, ℓ.Prime → ∃ x : ℤ, ¬ (ℓ : ℤ) ∣ (l.map fun ab => ab.1 * (x : ℤ) + ab.2).prod) ∧
    ∀ x : ℤ, (∀ ab ∈ l, 0 < ab.1 * x + ab.2 ∧ (ab.1 * x + ab.2).natAbs.Prime) →
      IsDCarmichael D (l.map fun ab => (ab.1 * x + ab.2).natAbs).prod ∧
        ((l.map fun ab => (ab.1 * x + ab.2).natAbs).prod.primeFactorsList.map
          fun p => jacobiSym D p) = σ

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Section 1 (definition of a universal D-form).

Uses 1 · Used by 4

Statement uses\(D\)-Carmichael number

Used byChernick-type universal forms for three prime factors · \(x(x+1)\) is never a universal form · Two-prime-factor \(D\)-Carmichael numbers · Universal \(5\)- and \(13\)-forms from prime quadruples

Theorem\(x(x+1)\) is never a universal form#✓fully proved

For every integer \(D\) and every finite sequence \(\sigma\) of integers, the product of the linear factors \(x\) and \(x + 1\) is not a universal \(D\)-form with respect to \(\sigma\): since \(x(x+1)\) is even for every integer \(x\), the prime \(2\) divides every value of the product.

theorem not_isUniversalLinearDForm_mul_succ (D : ℤ) (σ : List ℤ) :
    ¬ IsUniversalLinearDForm D σ [(1, 0), (1, 1)]

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Section 1 (definition of a universal D-form).

Uses 2

Statement usesUniversal \(D\)-forms

Proof uses\(D\)-Carmichael number

TheoremWilliams' Korselt criterion for \(D\)-Carmichael numbers#●ready to prove

Let \(D\) be an integer with \(D \equiv 0\) or \(1 \pmod 4\), and let \(n > 1\) be an odd composite natural number coprime to \(D\). Then \(n\) is a \(D\)-Carmichael number if and only if \(n\) is squarefree and \(p - \left(\frac{D}{p}\right)\) divides \(n - \left(\frac{D}{n}\right)\) for every prime \(p \mid n\), where \(\left(\frac{D}{m}\right)\) is the Jacobi symbol.

Primes and \(1\) satisfy the condition on the right but are not \(D\)-Carmichael, which is why \(n\) is assumed composite; it is assumed coprime to \(D\) as in the definition.

public theorem_wanted isDCarmichael_iff_korselt {D : ℤ} (hD : D % 4 = 0 ∨ D % 4 = 1) {n : ℕ}
    (hodd : Odd n) (hn : 1 < n) (hnp : ¬ n.Prime) (hDn : IsCoprime D n) :
    IsDCarmichael D n ↔
      Squarefree n ∧ ∀ p ∈ n.primeFactors, ((p : ℤ) - jacobiSym D p) ∣ ((n : ℤ) - jacobiSym D n)

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 1.

Uses 1

Statement uses\(D\)-Carmichael number

TheoremPrime \(k\)-tuples give infinitely many primary Carmichael numbers (Wagstaff)#●ready to prove

Suppose the prime \(k\)-tuples conjecture holds in the following form: for every finite list of integer pairs \((a_1, b_1), \dots, (a_k, b_k)\) with all \(a_i \ge 1\) such that for every prime \(\ell\) there is an integer \(x\) with \(\ell \nmid \prod_i (a_i x + b_i)\), there are infinitely many natural numbers \(x\) for which every \(a_i x + b_i\) is a positive prime. Then there are infinitely many composite natural numbers \(n > 1\) such that \(a^{n-1} \equiv 1 \pmod n\) for every natural number \(a\) coprime to \(n\), \(n\) has exactly three distinct prime factors, and for every prime \(p \mid n\) the sum of the base-\(p\) digits of \(n\) equals \(p\).

public theorem_wanted infinite_primary_carmichael_of_primeKTuples
    (hk : ∀ l : List (ℤ × ℤ), (∀ ab ∈ l, 1 ≤ ab.1) →
      (∀ ℓ : ℕ, ℓ.Prime → ∃ x : ℤ, ¬ (ℓ : ℤ) ∣ (l.map fun ab => ab.1 * x + ab.2).prod) →
      {x : ℕ | ∀ ab ∈ l,
        0 < ab.1 * (x : ℤ) + ab.2 ∧ (ab.1 * (x : ℤ) + ab.2).natAbs.Prime}.Infinite) :
    {n : ℕ | 1 < n ∧ ¬ n.Prime ∧ (∀ a : ℕ, a.Coprime n → a ^ (n - 1) ≡ 1 [MOD n]) ∧
      n.primeFactors.card = 3 ∧ ∀ p ∈ n.primeFactors, (Nat.digits p n).sum = p}.Infinite

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 2.

TheoremKorselt criterion for \(\sigma\)-Carmichael numbers#●ready to prove

Let \(\sigma = (\varepsilon_1, \dots, \varepsilon_r)\) be a list of signs \(\varepsilon_i \in \{1, -1\}\) of length \(r \ge 2\), put \(\varepsilon_\sigma = \varepsilon_1 \cdots \varepsilon_r\), and let \(n\) be an odd natural number. Then \(n\) is \(\sigma\)-Carmichael if and only if \(n\) is squarefree, has exactly \(r\) prime factors \(p_1 < \dots < p_r\), and satisfies \(p_i - \varepsilon_i \mid n - \varepsilon_\sigma\) for every \(i\).

public theorem_wanted isSignatureCarmichael_iff {σ : List ℤ} (hσ : ∀ e ∈ σ, e = 1 ∨ e = -1)
    (hlen : 2 ≤ σ.length) {n : ℕ} (hodd : Odd n) :
    IsSignatureCarmichael σ n ↔
      Squarefree n ∧ n.primeFactorsList.length = σ.length ∧
        ∀ pe ∈ n.primeFactorsList.zip σ, ((pe.1 : ℤ) - pe.2) ∣ ((n : ℤ) - σ.prod)

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 3.

Uses 1

Statement usesCarmichael numbers of a signature

TheoremTwo-prime-factor \(D\)-Carmichael numbers#●ready to prove

Let \(D\) be an integer and assume the prime \(k\)-tuples conjecture (the paper's Conjecture 1) in the following form: for every finite list of linear forms \(a_i x + b_i\) with integers \(a_i \ge 1\) and \(b_i\) such that for every prime \(\ell\) there is an integer \(x\) with \(\ell \nmid \prod_i (a_i x + b_i)\), there are infinitely many natural numbers \(x\) at which every \(a_i x + b_i\) is a positive prime. Then, first, for all primes \(p < q\) such that \(pq\) is a \(D\)-Carmichael number: the product of the linear factors \(Dx + p\) and \(Dx + q\) is a universal \(D\)-form with respect to the signature \((-1, 1)\); \(q = p + 2\); for every integer \(x\) at which \(p + Dx\) and \(q + Dx\) are both positive primes, their product is a primary \(D\)-Carmichael number; and there are infinitely many primary \(D\)-Carmichael numbers \(n\) whose Jacobi symbols \((D \mid \cdot)\) at the prime factors of \(n\), listed in increasing order with multiplicity, are \(-1, 1\). Second, a natural number \(n\) is a \(5\)-Carmichael number with exactly two prime factors counted with multiplicity if and only if \(n = (10x + 7)(10x + 9)\) for some natural number \(x \ge 1\) with \(10x + 7\) and \(10x + 9\) both prime.

public theorem_wanted twoFactor_isDCarmichael {D : ℤ}
    (hk : ∀ l : List (ℤ × ℤ), (∀ ab ∈ l, 1 ≤ ab.1) →
      (∀ ℓ : ℕ, ℓ.Prime → ∃ x : ℤ, ¬ (ℓ : ℤ) ∣ (l.map fun ab => ab.1 * x + ab.2).prod) →
      {x : ℕ | ∀ ab ∈ l,
        0 < ab.1 * (x : ℤ) + ab.2 ∧ (ab.1 * (x : ℤ) + ab.2).natAbs.Prime}.Infinite) :
    (∀ p q : ℕ, p.Prime → q.Prime → p < q → IsDCarmichael D (p * q) →
      IsUniversalLinearDForm D [-1, 1] [(D, p), (D, q)] ∧ q = p + 2 ∧
        (∀ x : ℤ, 0 < (p : ℤ) + D * x → ((p : ℤ) + D * x).natAbs.Prime →
          0 < (q : ℤ) + D * x → ((q : ℤ) + D * x).natAbs.Prime →
          IsPrimaryDCarmichael D (((p : ℤ) + D * x) * ((q : ℤ) + D * x)).natAbs) ∧
        {n : ℕ | IsPrimaryDCarmichael D n ∧
          n.primeFactorsList.map (fun r => jacobiSym D r) = [-1, 1]}.Infinite) ∧
    ∀ n : ℕ, (IsDCarmichael 5 n ∧ n.primeFactorsList.length = 2) ↔
      ∃ x : ℕ, 1 ≤ x ∧ (10 * x + 7).Prime ∧ (10 * x + 9).Prime ∧ n = (10 * x + 7) * (10 * x + 9)

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 4.

Uses 3

Statement uses\(D\)-Carmichael number · Primary \(D\)-Carmichael number · Universal \(D\)-forms

TheoremChernick-type universal forms for three prime factors#●ready to prove

Let \(D\) be an integer and let \(3 < p_1 < p_2 < p_3\) be primes such that \(p_1 p_2 p_3\) is \(D\)-Carmichael, and write \(\varepsilon_i = \left(\frac{D}{p_i}\right)\) for the Jacobi symbols. Let \(g\) be the greatest common divisor of \(p_1 - \varepsilon_1\), \(p_2 - \varepsilon_2\) and \(p_3 - \varepsilon_3\), and let \(c_1, c_2, c_3\) be pairwise coprime integers with \(g c_i = p_i - \varepsilon_i\) for each \(i\). Then

\[(D c_1^2 c_2 c_3 x + p_1)(D c_1 c_2^2 c_3 x + p_2)(D c_1 c_2 c_3^2 x + p_3)\]

is a universal \(D\)-form with respect to the signature \((\varepsilon_1, \varepsilon_2, \varepsilon_3)\).

Being \(D\)-Carmichael includes the condition \(D \equiv 0\) or \(1 \pmod 4\).

public theorem_wanted isUniversalLinearDForm_chernick {D : ℤ} {p₁ p₂ p₃ : ℕ}
    (h₁ : p₁.Prime) (h₂ : p₂.Prime) (h₃ : p₃.Prime) (h3 : 3 < p₁)
    (h12 : p₁ < p₂) (h23 : p₂ < p₃) (hC : IsDCarmichael D (p₁ * p₂ * p₃))
    {g c₁ c₂ c₃ : ℤ}
    (hg : g = Int.gcd (Int.gcd (p₁ - jacobiSym D p₁) (p₂ - jacobiSym D p₂))
      (p₃ - jacobiSym D p₃))
    (hc₁ : g * c₁ = p₁ - jacobiSym D p₁) (hc₂ : g * c₂ = p₂ - jacobiSym D p₂)
    (hc₃ : g * c₃ = p₃ - jacobiSym D p₃)
    (hc₁₂ : IsCoprime c₁ c₂) (hc₁₃ : IsCoprime c₁ c₃) (hc₂₃ : IsCoprime c₂ c₃) :
    IsUniversalLinearDForm D [jacobiSym D p₁, jacobiSym D p₂, jacobiSym D p₃]
      [(D * c₁ ^ 2 * c₂ * c₃, p₁), (D * c₁ * c₂ ^ 2 * c₃, p₂), (D * c₁ * c₂ * c₃ ^ 2, p₃)]

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 5.

Uses 2

Statement uses\(D\)-Carmichael number · Universal \(D\)-forms

TheoremPrime \(k\)-tuples give primary \(5\)-Carmichael numbers of every three-term signature#●ready to prove

Suppose the prime \(k\)-tuples conjecture holds in the following form: for every finite list of integer pairs \((a_1, b_1), \dots, (a_k, b_k)\) with all \(a_i \ge 1\) such that for every prime \(\ell\) there is an integer \(x\) with \(\ell \nmid \prod_i (a_i x + b_i)\), there are infinitely many natural numbers \(x\) for which every \(a_i x + b_i\) is a positive prime. Then for all signs \(\sigma_1, \sigma_2, \sigma_3 \in \{1, -1\}\) there are infinitely many primary \(5\)-Carmichael numbers \(n\) that have exactly three prime factors \(p_1 \le p_2 \le p_3\) counted with multiplicity and satisfy \(\left(\frac{5}{p_i}\right) = \sigma_i\) for \(i = 1, 2, 3\), where \(\left(\frac{5}{p}\right)\) is the Jacobi symbol.

public theorem_wanted infinite_primary_fiveCarmichael_of_primeKTuples
    (hk : ∀ l : List (ℤ × ℤ), (∀ ab ∈ l, 1 ≤ ab.1) →
      (∀ ℓ : ℕ, ℓ.Prime → ∃ x : ℤ, ¬ (ℓ : ℤ) ∣ (l.map fun ab => ab.1 * x + ab.2).prod) →
      {x : ℕ | ∀ ab ∈ l,
        0 < ab.1 * (x : ℤ) + ab.2 ∧ (ab.1 * (x : ℤ) + ab.2).natAbs.Prime}.Infinite)
    (σ : List ℤ) (hσ : ∀ e ∈ σ, e = 1 ∨ e = -1) (hlen : σ.length = 3) :
    {n : ℕ | IsPrimaryDCarmichael 5 n ∧
      n.primeFactorsList.map (fun p => jacobiSym 5 p) = σ}.Infinite

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 6.

Uses 1

Statement usesPrimary \(D\)-Carmichael number

TheoremThe \(D\)-Carmichael numbers of the form \(3qr\)#●ready to prove

Let \(q < r\) be primes with \(3 < q\). Then \(3qr\) is a \(D\)-Carmichael number for some integer \(D\) if and only if \(3qr\) is one of \(105\), \(195\), \(255\), \(399\), \(561\) and \(2703\). Moreover, for all integers \(D\) and \(D'\) such that \(3qr\) is both \(D\)-Carmichael and \(D'\)-Carmichael, the Jacobi symbols agree at \(3\), \(q\) and \(r\): \((D \mid 3) = (D' \mid 3)\), \((D \mid q) = (D' \mid q)\) and \((D \mid r) = (D' \mid r)\).

public theorem_wanted threeQR_isDCarmichael_iff {q r : ℕ} (hq : q.Prime) (hr : r.Prime)
    (h3q : 3 < q) (hqr : q < r) :
    ((∃ D : ℤ, IsDCarmichael D (3 * q * r)) ↔
      3 * q * r ∈ ({105, 195, 255, 399, 561, 2703} : Finset ℕ)) ∧
    ∀ D D' : ℤ, IsDCarmichael D (3 * q * r) → IsDCarmichael D' (3 * q * r) →
      jacobiSym D 3 = jacobiSym D' 3 ∧ jacobiSym D q = jacobiSym D' q ∧
        jacobiSym D r = jacobiSym D' r

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 7.

Uses 1

Statement uses\(D\)-Carmichael number

TheoremPrimary and secondary \(5\)-Carmichael numbers from prime quadruples#●ready to prove

Assume the prime \(k\)-tuples conjecture (the paper's Conjecture 1) in the following form: for every finite list of linear forms \(a_i x + b_i\) with integers \(a_i \ge 1\) and \(b_i\) such that for every prime \(\ell\) there is an integer \(x\) with \(\ell \nmid \prod_i (a_i x + b_i)\), there are infinitely many natural numbers \(x\) at which every \(a_i x + b_i\) is a positive prime. For a natural number \(x\) put \(p = 30x + 17\), \(q = 30x + 19\), \(r = 180x + 107\) and \(s = 180x + 109\). Then, first, for every natural number \(x\) with \(p\), \(q\), \(r\) and \(s\) all prime, \(pqr\) and \(pqs\) are primary \(5\)-Carmichael numbers and \(pqrs\) is a secondary \(5\)-Carmichael number; and second, there are infinitely many natural numbers \(x\) with \(p\), \(q\), \(r\) and \(s\) all prime.

public theorem_wanted primaryFiveCarmichael_pairs_of_primeKTuples
    (hk : ∀ l : List (ℤ × ℤ), (∀ ab ∈ l, 1 ≤ ab.1) →
      (∀ ℓ : ℕ, ℓ.Prime → ∃ x : ℤ, ¬ (ℓ : ℤ) ∣ (l.map fun ab => ab.1 * x + ab.2).prod) →
      {x : ℕ | ∀ ab ∈ l,
        0 < ab.1 * (x : ℤ) + ab.2 ∧ (ab.1 * (x : ℤ) + ab.2).natAbs.Prime}.Infinite) :
    (∀ x : ℕ, (30 * x + 17).Prime → (30 * x + 19).Prime → (180 * x + 107).Prime →
      (180 * x + 109).Prime →
      IsPrimaryDCarmichael 5 ((30 * x + 17) * (30 * x + 19) * (180 * x + 107)) ∧
      IsPrimaryDCarmichael 5 ((30 * x + 17) * (30 * x + 19) * (180 * x + 109)) ∧
      IsSecondaryDCarmichael 5
        ((30 * x + 17) * (30 * x + 19) * (180 * x + 107) * (180 * x + 109))) ∧
    {x : ℕ | (30 * x + 17).Prime ∧ (30 * x + 19).Prime ∧ (180 * x + 107).Prime ∧
      (180 * x + 109).Prime}.Infinite

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 8.

Uses 2

Statement usesPrimary \(D\)-Carmichael number · Secondary \(D\)-Carmichael number

TheoremPrimary and secondary \(13\)-Carmichael numbers from prime quadruples#●ready to prove

Assume the prime \(k\)-tuples conjecture (the paper's Conjecture 1) in the following form: for every finite list of linear forms \(a_i x + b_i\) with integers \(a_i \ge 1\) and \(b_i\) such that for every prime \(\ell\) there is an integer \(x\) with \(\ell \nmid \prod_i (a_i x + b_i)\), there are infinitely many natural numbers \(x\) at which every \(a_i x + b_i\) is a positive prime. For a natural number \(x\) put \(p = 78x + 137\), \(q = 78x + 139\), \(r = 468x + 827\) and \(s = 468x + 829\). Then, first, for every natural number \(x\) with \(p\), \(q\), \(r\) and \(s\) all prime: \(pqr\) and \(pqs\) are primary \(13\)-Carmichael numbers; the Jacobi symbols \((13 \mid \cdot)\) at the prime factors of \(pqr\), listed in increasing order, are \(-1, 1, -1\), and those at the prime factors of \(pqs\) are \(-1, 1, 1\); and \(pqrs\) is a secondary \(13\)-Carmichael number. Second, there are infinitely many natural numbers \(x\) with \(p\), \(q\), \(r\) and \(s\) all prime.

MetaMathlibExt.primaryThirteenCarmichael_pairs_of_primeKTuples · 17 lines

public theorem_wanted primaryThirteenCarmichael_pairs_of_primeKTuples
    (hk : ∀ l : List (ℤ × ℤ), (∀ ab ∈ l, 1 ≤ ab.1) →
      (∀ ℓ : ℕ, ℓ.Prime → ∃ x : ℤ, ¬ (ℓ : ℤ) ∣ (l.map fun ab => ab.1 * x + ab.2).prod) →
      {x : ℕ | ∀ ab ∈ l,
        0 < ab.1 * (x : ℤ) + ab.2 ∧ (ab.1 * (x : ℤ) + ab.2).natAbs.Prime}.Infinite) :
    (∀ x : ℕ, (78 * x + 137).Prime → (78 * x + 139).Prime → (468 * x + 827).Prime →
      (468 * x + 829).Prime →
      IsPrimaryDCarmichael 13 ((78 * x + 137) * (78 * x + 139) * (468 * x + 827)) ∧
      IsPrimaryDCarmichael 13 ((78 * x + 137) * (78 * x + 139) * (468 * x + 829)) ∧
      (((78 * x + 137) * (78 * x + 139) * (468 * x + 827)).primeFactorsList.map
        fun p => jacobiSym 13 p) = [-1, 1, -1] ∧
      (((78 * x + 137) * (78 * x + 139) * (468 * x + 829)).primeFactorsList.map
        fun p => jacobiSym 13 p) = [-1, 1, 1] ∧
      IsSecondaryDCarmichael 13
        ((78 * x + 137) * (78 * x + 139) * (468 * x + 827) * (468 * x + 829))) ∧
    {x : ℕ | (78 * x + 137).Prime ∧ (78 * x + 139).Prime ∧ (468 * x + 827).Prime ∧
      (468 * x + 829).Prime}.Infinite

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 9.

Uses 2

Statement usesPrimary \(D\)-Carmichael number · Secondary \(D\)-Carmichael number

TheoremUniversal \(5\)- and \(13\)-forms from prime quadruples#●ready to prove

Assume the prime \(k\)-tuples conjecture (the paper's Conjecture 1) in the following form: for every finite list of linear forms \(a_i x + b_i\) with integers \(a_i \ge 1\) and \(b_i\) such that for every prime \(\ell\) there is an integer \(x\) with \(\ell \nmid \prod_i (a_i x + b_i)\), there are infinitely many natural numbers \(x\) at which every \(a_i x + b_i\) is a positive prime. Then, first, for \(D = 5\) and for \(D = 13\): the products of linear factors \((390z + 137)(390z + 139)(2340z + 827)\) and \((390z + 137)(390z + 139)(2340z + 829)\) are universal \(D\)-forms with respect to the signatures \((-1, 1, -1)\) and \((-1, 1, 1)\) respectively; the product \((390z + 137)(390z + 139)(2340z + 827)(2340z + 829)\) is a universal \(D\)-form with respect to the signature \((-1, 1, -1, 1)\); and for every natural number \(z\) with \(390z + 137\) and \(390z + 139\) prime, the first product is a primary \(D\)-Carmichael number if \(2340z + 827\) is prime, the second is a primary \(D\)-Carmichael number if \(2340z + 829\) is prime, and the four-factor product is a secondary \(D\)-Carmichael number if both are prime. Second, there are infinitely many natural numbers \(z\) at which all four factors are prime.

The paper (and the record) print "universal 7- and 13-forms"; \(7 \not\equiv 0, 1 \pmod 4\) is not a discriminant, and the next sentence and the proof (combining the \(5\)-form of Theorem 8 with the \(13\)-form of Theorem 9) fix it as \(5\).

MetaMathlibExt.universal_fiveThirteen_forms · 20 lines

public theorem_wanted universal_fiveThirteen_forms
    (hk : ∀ l : List (ℤ × ℤ), (∀ ab ∈ l, 1 ≤ ab.1) →
      (∀ ℓ : ℕ, ℓ.Prime → ∃ x : ℤ, ¬ (ℓ : ℤ) ∣ (l.map fun ab => ab.1 * x + ab.2).prod) →
      {x : ℕ | ∀ ab ∈ l,
        0 < ab.1 * (x : ℤ) + ab.2 ∧ (ab.1 * (x : ℤ) + ab.2).natAbs.Prime}.Infinite) :
    (∀ D ∈ ({5, 13} : Finset ℤ),
      IsUniversalLinearDForm D [-1, 1, -1] [(390, 137), (390, 139), (2340, 827)] ∧
      IsUniversalLinearDForm D [-1, 1, 1] [(390, 137), (390, 139), (2340, 829)] ∧
      IsUniversalLinearDForm D [-1, 1, -1, 1]
        [(390, 137), (390, 139), (2340, 827), (2340, 829)] ∧
      ∀ z : ℕ, (390 * z + 137).Prime → (390 * z + 139).Prime →
        ((2340 * z + 827).Prime →
          IsPrimaryDCarmichael D ((390 * z + 137) * (390 * z + 139) * (2340 * z + 827))) ∧
        ((2340 * z + 829).Prime →
          IsPrimaryDCarmichael D ((390 * z + 137) * (390 * z + 139) * (2340 * z + 829))) ∧
        ((2340 * z + 827).Prime → (2340 * z + 829).Prime →
          IsSecondaryDCarmichael D
            ((390 * z + 137) * (390 * z + 139) * (2340 * z + 827) * (2340 * z + 829)))) ∧
    {z : ℕ | (390 * z + 137).Prime ∧ (390 * z + 139).Prime ∧ (2340 * z + 827).Prime ∧
      (2340 * z + 829).Prime}.Infinite

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 10.

Uses 3

Statement usesPrimary \(D\)-Carmichael number · Secondary \(D\)-Carmichael number · Universal \(D\)-forms

TheoremSchinzel's Hypothesis H gives numbers that are primary \(5\)- and \((-11)\)-Carmichael#●ready to prove

Suppose Schinzel's Hypothesis H holds in the following form: for every finite set \(F\) of integer polynomials, each satisfying the Bunyakovsky condition, such that \(F\) satisfies the Schinzel condition, there are infinitely many natural numbers \(n\) for which \(|f(n)|\) is prime for every \(f \in F\). Then there are infinitely many natural numbers \(n\) that are both primary \(5\)-Carmichael and primary \((-11)\)-Carmichael and have exactly three prime factors \(p_1 \le p_2 \le p_3\) counted with multiplicity, with Jacobi symbols \(\left(\frac{5}{p_1}\right) = 1\), \(\left(\frac{5}{p_2}\right) = \left(\frac{5}{p_3}\right) = -1\), \(\left(\frac{-11}{p_1}\right) = \left(\frac{-11}{p_3}\right) = -1\) and \(\left(\frac{-11}{p_2}\right) = 1\).

public theorem_wanted infinite_primary_five_negEleven_of_schinzelH
    (hH : ∀ fs : Finset (Polynomial ℤ), (∀ f ∈ fs, Polynomial.BunyakovskyCondition f) →
      Polynomial.SchinzelCondition fs →
      {n : ℕ | ∀ f ∈ fs, (f.eval (n : ℤ)).natAbs.Prime}.Infinite) :
    {n : ℕ | IsPrimaryDCarmichael 5 n ∧
      n.primeFactorsList.map (fun p => jacobiSym 5 p) = [1, -1, -1] ∧
      IsPrimaryDCarmichael (-11) n ∧
      n.primeFactorsList.map (fun p => jacobiSym (-11) p) = [-1, 1, -1]}.Infinite

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 11.

Uses 3

Statement usesPrimary \(D\)-Carmichael number · Bunyakovsky condition · Schinzel condition

TheoremBi-Carmichael numbers for signatures of opposite signs#●ready to prove

Let \(D\) and \(D'\) be integers and let \(n\) be both a \(D\)-Carmichael and a \(D'\)-Carmichael number whose Jacobi symbols satisfy \(\left(\frac{D}{n}\right) \ne \left(\frac{D'}{n}\right)\). Then \(\left(\frac{D'}{p}\right) = -\left(\frac{D}{p}\right)\) for every prime \(p \mid n\), the number of distinct prime factors of \(n\) is odd, and \(3 \nmid n\).

This reads the paper's \(n \in C(\sigma) \cap C(\sigma')\) as \(n\) being \(D\)-Carmichael with signature \(\sigma\) and \(D'\)-Carmichael with signature \(\sigma'\).

public theorem_wanted isDCarmichael_opposite_signs {D D' : ℤ} {n : ℕ} (h : IsDCarmichael D n)
    (h' : IsDCarmichael D' n)
    (hs : jacobiSym D n ≠ jacobiSym D' n) :
    (∀ p ∈ n.primeFactors, jacobiSym D' p = -jacobiSym D p) ∧ Odd n.primeFactors.card ∧
      ¬ 3 ∣ n

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 12.

Uses 1

Statement uses\(D\)-Carmichael number

TheoremThree-prime-factor \(D\)-Carmichael numbers have a single Jacobi sign#●ready to prove

Let \(D\) and \(D'\) be integers and let \(n\) be a natural number with exactly three distinct prime factors. If \(n\) is both \(D\)-Carmichael and \(D'\)-Carmichael, then the Jacobi symbols agree: \(\left(\frac{D}{n}\right) = \left(\frac{D'}{n}\right)\).

public theorem_wanted jacobiSym_eq_of_isDCarmichael_three {D D' : ℤ} {n : ℕ}
    (h3 : n.primeFactors.card = 3) (h : IsDCarmichael D n) (h' : IsDCarmichael D' n) :
    jacobiSym D n = jacobiSym D' n

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Theorem 13.

Uses 1

Statement uses\(D\)-Carmichael number

TheoremBallot's Lemma 3: the three divisibilities persist modulo \(c_1 c_2 c_3\)#●ready to prove

Let \(\varepsilon_1, \varepsilon_2, \varepsilon_3 \in \{1, -1\}\), let \(c_1, c_2, c_3\) be pairwise coprime positive integers, let \(\varepsilon = \varepsilon_1 \varepsilon_2 \varepsilon_3\), and let \(g \ne 0\) be an integer such that \(c_i g\) divides \((c_1 g + \varepsilon_1)(c_2 g + \varepsilon_2)(c_3 g + \varepsilon_3) - \varepsilon\) for \(i = 1, 2, 3\). Then every nonzero integer \(h \equiv g \pmod{c_1 c_2 c_3}\) satisfies the same divisibilities: \(c_i h\) divides \((c_1 h + \varepsilon_1)(c_2 h + \varepsilon_2)(c_3 h + \varepsilon_3) - \varepsilon\) for \(i = 1, 2, 3\).

The divisibility hypothesis on \(g\) is needed: for \((c_1, c_2, c_3) = (1, 2, 3)\), \(\varepsilon_1 = \varepsilon_2 = \varepsilon_3 = 1\) and \(g = h = 1\), the product is \(24\) and \(2 \nmid 24 - 1\).

public theorem_wanted mul_dvd_prod_sub_sign_of_modEq {ε₁ ε₂ ε₃ c₁ c₂ c₃ g h : ℤ}
    (hε₁ : ε₁ = 1 ∨ ε₁ = -1) (hε₂ : ε₂ = 1 ∨ ε₂ = -1) (hε₃ : ε₃ = 1 ∨ ε₃ = -1)
    (hc₁ : 0 < c₁) (hc₂ : 0 < c₂) (hc₃ : 0 < c₃)
    (h₁₂ : IsCoprime c₁ c₂) (h₁₃ : IsCoprime c₁ c₃) (h₂₃ : IsCoprime c₂ c₃) (hg : g ≠ 0)
    (hgdvd : ∀ c ∈ ({c₁, c₂, c₃} : Finset ℤ),
      c * g ∣ (c₁ * g + ε₁) * (c₂ * g + ε₂) * (c₃ * g + ε₃) - ε₁ * ε₂ * ε₃)
    (hh : h ≠ 0) (hhg : h ≡ g [ZMOD c₁ * c₂ * c₃]) :
    ∀ c ∈ ({c₁, c₂, c₃} : Finset ℤ),
      c * h ∣ (c₁ * h + ε₁) * (c₂ * h + ε₂) * (c₃ * h + ε₃) - ε₁ * ε₂ * ε₃

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Lemma 3.

TheoremDivisibility conditions for \(D\)-Carmichael numbers \(3qr\)#●ready to prove

Let \(D\) be an integer, let \(q < r\) be primes with \(3 < q\), and write \(\alpha = (D \mid 3)\), \(\beta = (D \mid q)\) and \(\gamma = (D \mid r)\) for the Jacobi symbols. If \(3qr\) is a \(D\)-Carmichael number, then \(q - \beta\) divides \(3r - \alpha\gamma\) and \(r - \gamma\) divides \(3q - \alpha\beta\); moreover \(r = 3q + 2\gamma\) if \(\gamma = -\alpha\beta\), and \(2r = 3q + \gamma\) if \(\gamma = \alpha\beta\).

public theorem_wanted threeQR_isDCarmichael_divisibility {D : ℤ} {q r : ℕ} (hq : q.Prime)
    (hr : r.Prime) (h3q : 3 < q) (hqr : q < r)
    (h : IsDCarmichael D (3 * q * r)) :
    ((q : ℤ) - jacobiSym D q) ∣ (3 * r - jacobiSym D 3 * jacobiSym D r) ∧
    ((r : ℤ) - jacobiSym D r) ∣ (3 * q - jacobiSym D 3 * jacobiSym D q) ∧
    (jacobiSym D r = -(jacobiSym D 3 * jacobiSym D q) → (r : ℤ) = 3 * q + 2 * jacobiSym D r) ∧
    (jacobiSym D r = jacobiSym D 3 * jacobiSym D q → 2 * (r : ℤ) = 3 * q + jacobiSym D r)

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Lemma 4.

Uses 1

Statement uses\(D\)-Carmichael number

TheoremWilliams' characterization of two-prime \(D\)-Carmichael numbers#●ready to prove

Let \(D\) be an integer with \(D \equiv 0\) or \(1 \pmod 4\) and let \(p_1 < p_2\) be primes. Then \(p_1 p_2\) is a \(D\)-Carmichael number if and only if \(p_2 = p_1 + 2\), \(\left(\frac{D}{p_1}\right) = -1\) and \(\left(\frac{D}{p_2}\right) = 1\), where \(\left(\frac{D}{p}\right)\) is the Jacobi symbol.

public theorem_wanted isDCarmichael_mul_iff {D : ℤ} (hD : D % 4 = 0 ∨ D % 4 = 1) {p₁ p₂ : ℕ}
    (h₁ : p₁.Prime) (h₂ : p₂.Prime) (h12 : p₁ < p₂) :
    IsDCarmichael D (p₁ * p₂) ↔ p₂ = p₁ + 2 ∧ jacobiSym D p₁ = -1 ∧ jacobiSym D p₂ = 1

Sources

• Christian Ballot, On Absolute Lucas Pseudoprimes, INTEGERS 26 (2026), Article A102, Proposition 1.

Uses 1

Statement uses\(D\)-Carmichael number

12 fully proved18 ready to prove

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