Three-term sums with prescribed square classes cover \(\mathbb{Z}_{p^r}\) to Consecutive zero-sum constant of a finite abelian group

TheoremThree-term sums with prescribed square classes cover \(\mathbb{Z}_{p^r}\)#●ready to prove

Let \(p \ge 7\) be a prime, \(r \ge 1\) and \(n = p^r\); write \(\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}\), \(U(n)\) for its group of units, \(A_1 = U(n)^2\) for the set of squares of units and \(A_2 = U(n) \setminus U(n)^2\). For all units \(x_1, x_2, x_3\) of \(\mathbb{Z}_n\) and every \(f : \{1, 2, 3\} \to \{1, 2\}\), \(A_{f(1)} x_1 + A_{f(2)} x_2 + A_{f(3)} x_3 = \mathbb{Z}_n\): every \(z \in \mathbb{Z}_n\) equals \(a_1 x_1 + a_2 x_2 + a_3 x_3\) for some \(a_1 \in A_{f(1)}\), \(a_2 \in A_{f(2)}\), \(a_3 \in A_{f(3)}\).

The hypothesis \(r \ge 1\) is needed: for \(r = 0\), \(\mathbb{Z}_1 = \{0\}\) has \(0 = 0^2\) as its only unit, so \(A_2\) is empty.

public theorem_wanted unit_square_class_three_sumset_eq_univ (p r : ℕ) (hp : p.Prime)
    (h7 : 7 ≤ p) (hr : 0 < r) (x : Fin 3 → ZMod (p ^ r)) (hx : ∀ i, IsUnit (x i))
    (f : Fin 3 → Bool) (z : ZMod (p ^ r)) :
    ∃ a : Fin 3 → ZMod (p ^ r),
      (∀ i, IsUnit (a i) ∧ (a i ∈ unitSqSet (p ^ r) ↔ f i = true)) ∧ ∑ i, a i * x i = z

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Lemma 18 (jis-2024-24.1.7-lem-18).

TheoremThree unit terms force a square-weighted zero sum modulo \(p^r\)#●ready to prove

Let \(p \ge 7\) be a prime, \(r\) a natural number and \(n = p^r\), and let \(U(n)^2\) be the set of squares of units of \(\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}\). If a sequence \((x_1, \dots, x_k)\) in \(\mathbb{Z}_n\) has at least three indices \(i\) with \(x_i\) a unit, then there are \(a_1, \dots, a_k \in U(n)^2\) with \(a_1 x_1 + \dots + a_k x_k = 0\), that is, the sequence is a \(U(n)^2\)-weighted zero-sum sequence.

public theorem_wanted unit_square_weighted_zero_sum_prime_power (p r k : ℕ) (hp : p.Prime)
    (h7 : 7 ≤ p) (x : Fin k → ZMod (p ^ r))
    (h : 3 ≤ (Finset.univ.filter fun i => IsUnit (x i)).card) :
    IsWeightedZeroSum (p ^ r) k (unitSqSet (p ^ r)) x

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Corollary 19 (jis-2024-24.1.7-cor-19).

TheoremFew unit terms force a Jacobi-weighted zero sum#●ready to prove

Let \(n\) be an odd squarefree natural number, \(\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}\), and \(S(n)\) the set of units \(u\) of \(\mathbb{Z}_n\) whose Jacobi symbol \((a/n)\) is \(1\), where \(0 \le a < n\) is the representative of \(u\). Let \((x_1, \dots, x_k)\) be a sequence in \(\mathbb{Z}_n\), and call \(x_i\) coprime to a prime \(p\) when its representative in \(\{0, \dots, n-1\}\) is. Suppose that for every prime divisor \(p\) of \(n\) at least two indices \(i\) have \(x_i\) coprime to \(p\), and that at most one index \(i\) has \(x_i\) a unit. Then there are \(a_1, \dots, a_k \in S(n)\) with \(a_1 x_1 + \dots + a_k x_k = 0\), that is, the sequence is an \(S(n)\)-weighted zero-sum sequence.

public theorem_wanted jacobi_weighted_zero_sum_of_few_units (n k : ℕ) (hn : Odd n)
    (hsf : Squarefree n) (x : Fin k → ZMod n)
    (h2 : ∀ p ∈ n.primeFactors, 2 ≤ (Finset.univ.filter fun i => Nat.Coprime (x i).val p).card)
    (h1 : (Finset.univ.filter fun i => IsUnit (x i)).card ≤ 1) :
    IsWeightedZeroSum n k (jacobiOneSet n) x

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Lemma 21 (jis-2024-24.1.7-lem-21).

TheoremThree terms coprime to one prime force a Jacobi-weighted zero sum#●ready to prove

Let \(n\) be a squarefree natural number all of whose prime divisors are at least \(7\), let \(\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}\), and let \(S(n)\) be the set of units \(u\) of \(\mathbb{Z}_n\) whose Jacobi symbol \((a/n)\) is \(1\), where \(0 \le a < n\) is the representative of \(u\). Let \((x_1, \dots, x_k)\) be a sequence in \(\mathbb{Z}_n\), and call \(x_i\) coprime to a prime \(p\) when its representative in \(\{0, \dots, n-1\}\) is. Suppose that for every prime divisor \(p\) of \(n\) at least two indices \(i\) have \(x_i\) coprime to \(p\), and that for some prime divisor \(p\) of \(n\) at least three indices \(i\) have \(x_i\) coprime to \(p\). Then there are \(a_1, \dots, a_k \in S(n)\) with \(a_1 x_1 + \dots + a_k x_k = 0\), that is, the sequence is an \(S(n)\)-weighted zero-sum sequence.

public theorem_wanted jacobi_weighted_zero_sum_of_three_coprime (n k : ℕ) (hsf : Squarefree n)
    (h7 : ∀ p ∈ n.primeFactors, 7 ≤ p) (x : Fin k → ZMod n)
    (h2 : ∀ p ∈ n.primeFactors, 2 ≤ (Finset.univ.filter fun i => Nat.Coprime (x i).val p).card)
    (h3 : ∃ p ∈ n.primeFactors, 3 ≤ (Finset.univ.filter fun i => Nat.Coprime (x i).val p).card) :
    IsWeightedZeroSum n k (jacobiOneSet n) x

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Lemma 22 (jis-2024-24.1.7-lem-22).

TheoremDavenport constant for the weights of Jacobi symbol one#●ready to prove

For a natural number \(n\) and \(A \subseteq \mathbb{Z}/n\mathbb{Z}\), let \(D_A(n)\) denote the least positive integer \(k\) such that every sequence \((x_1, \dots, x_k)\) in \(\mathbb{Z}/n\mathbb{Z}\) has a nonempty set \(J\) of indices and weights \(a_i \in A\) for \(i \in J\) with \(\sum_{i \in J} a_i x_i = 0\). Let \(n\) be an odd squarefree natural number and let \(S(n)\) be the set of units \(x\) of \(\mathbb{Z}/n\mathbb{Z}\) with Jacobi symbol \(\left(\frac{x}{n}\right) = 1\). If \(n\) is prime, then \(D_{S(n)}(n) = 3\). If \(n\) is not prime and every prime divisor of \(n\) is at least \(7\), then \(D_{S(n)}(n) = \Omega(n) + 1\), where \(\Omega(n)\) is the number of prime factors of \(n\) counted with multiplicity.

public theorem_wanted jacobi_davenport_constant (n : ℕ) (hn : Odd n) (hsf : Squarefree n) :
    (n.Prime → IsLeast (davenportSet n (jacobiOneSet n)) 3) ∧
    (¬ n.Prime → (∀ p ∈ n.primeFactors, 7 ≤ p) →
      IsLeast (davenportSet n (jacobiOneSet n)) (Ω n + 1))

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Theorem 23 (jis-2024-24.1.7-thm-23).

TheoremConsecutive constant for the weights of Jacobi symbol one#●ready to prove

For a natural number \(n\) and \(A \subseteq \mathbb{Z}/n\mathbb{Z}\), let \(C_A(n)\) denote the least positive integer \(k\) such that every sequence \((x_1, \dots, x_k)\) in \(\mathbb{Z}/n\mathbb{Z}\) has a block of consecutive terms \(x_s, \dots, x_t\) with \(s \le t\) and weights \(a_s, \dots, a_t \in A\) such that \(a_s x_s + \dots + a_t x_t = 0\). Let \(n\) be an odd squarefree natural number and let \(S(n)\) be the set of units \(x\) of \(\mathbb{Z}/n\mathbb{Z}\) with Jacobi symbol \(\left(\frac{x}{n}\right) = 1\). If \(n\) is prime, then \(C_{S(n)}(n) = 3\). If \(n\) is not prime and every prime divisor of \(n\) is at least \(7\), then \(C_{S(n)}(n) = 2^{\Omega(n)}\), where \(\Omega(n)\) is the number of prime factors of \(n\) counted with multiplicity.

public theorem_wanted jacobi_consecutive_constant (n : ℕ) (hn : Odd n) (hsf : Squarefree n) :
    (n.Prime → IsLeast (consecutiveSet n (jacobiOneSet n)) 3) ∧
    (¬ n.Prime → (∀ p ∈ n.primeFactors, 7 ≤ p) →
      IsLeast (consecutiveSet n (jacobiOneSet n)) (2 ^ Ω n))

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Theorem 24 (jis-2024-24.1.7-thm-24).

TheoremUnits with equal Jacobi and Legendre symbols form a subgroup of index two#●ready to prove

Let \(n\) be an odd natural number and \(p\) a prime divisor of \(n\), and suppose that \(p\) is not the unique prime divisor of \(n\) with odd exponent in \(n\); that is, it is not the case that the exponent of \(p\) in \(n\) is odd and every prime divisor of \(n\) with odd exponent in \(n\) equals \(p\). Then the set of units \(u\) of \(\mathbb{Z}/n\mathbb{Z}\) whose Jacobi symbol \(\left(\frac{u}{n}\right)\) equals the Legendre symbol \(\left(\frac{u}{p}\right)\) is a subgroup of index two in the unit group of \(\mathbb{Z}/n\mathbb{Z}\).

public theorem_wanted jacobi_eq_legendre_index_two (n p : ℕ) (hn : Odd n) (hp : p.Prime)
    (hpn : p ∣ n)
    (hnot : ¬ (Odd (n.factorization p) ∧
      ∀ q ∈ n.primeFactors, Odd (n.factorization q) → q = p)) :
    ∃ H : Subgroup (ZMod n)ˣ,
      (∀ u : (ZMod n)ˣ, u ∈ H ↔
        jacobiSym (((u : ZMod n).val : ℕ) : ℤ) n = jacobiSym (((u : ZMod n).val : ℕ) : ℤ) p) ∧
      H.index = 2

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Proposition 26 (jis-2024-24.1.7-prop-26).

TheoremLifting the Jacobi kernel into the Jacobi agreement set#●ready to prove

Let \(n\) be an odd natural number and let \(p\), \(p'\) be prime divisors of \(n\) with \(p\) coprime to \(n' = n/p\). For a residue \(x\) modulo \(N\) and an odd \(m\), write \((x/m)\) for the Jacobi symbol \((a/m)\) of the representative \(0 \le a < N\) of \(x\). Let \(U(m)\) be the group of units of \(\mathbb{Z}_m = \mathbb{Z}/m\mathbb{Z}\), let \(S(n') = \{y \in U(n') : (y/n') = 1\}\) and \(L(n;p') = \{x \in U(n) : (x/n) = (x/p')\}\), and let \(f_{n,n'} : \mathbb{Z}_n \to \mathbb{Z}_{n'}\) be reduction modulo \(n'\). Then \(S(n') \subseteq f_{n,n'}(L(n;p'))\): every \(y \in S(n')\) equals \(f_{n,n'}(x)\) for some \(x \in L(n;p')\).

public theorem_wanted jacobi_kernel_subset_image_jacobi_eq_legendre (n p p' : ℕ) (hn : Odd n)
    (hp : p.Prime) (hp' : p'.Prime) (hpn : p ∣ n) (hp'n : p' ∣ n)
    (hcop : Nat.Coprime p (n / p)) (y : ZMod (n / p)) (hy : y ∈ jacobiOneSet (n / p)) :
    ∃ x ∈ jacobiEqSet n p', ZMod.castHom (Nat.div_dvd_of_dvd hpn) (ZMod (n / p)) x = y

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Lemma 28 (jis-2024-24.1.7-lem-28).

TheoremUnits modulo a prime divisor lift to units where the Jacobi and Legendre symbols agree#●ready to prove

Let \(n\) be odd and let \(p'\) be a prime dividing \(n\) such that \(p'\) is coprime to \(n/p'\). Write \(L(n;p')\) for the set of units \(x\) of \(\mathbb{Z}/n\mathbb{Z}\) whose least nonnegative representative \(v\) satisfies \(\left(\frac{v}{n}\right) = \left(\frac{v}{p'}\right)\), where both are Jacobi symbols. Then every unit \(z\) of \(\mathbb{Z}/p'\mathbb{Z}\) is the reduction modulo \(p'\) of some element of \(L(n;p')\).

public theorem_wanted units_prime_subset_image_jacobi_eq_legendre (n p' : ℕ) (hn : Odd n)
    (hp' : p'.Prime) (hp'n : p' ∣ n) (hcop : Nat.Coprime p' (n / p')) (z : ZMod p')
    (hz : IsUnit z) :
    ∃ x ∈ jacobiEqSet n p', ZMod.castHom hp'n (ZMod p') x = z

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Lemma 29 (jis-2024-24.1.7-lem-29).

TheoremChinese remainder lifts into the Jacobi agreement set#●ready to prove

Let \(n\) be an odd squarefree natural number, \(p'\) a prime divisor of \(n\) and \(n' = n/p'\). For a residue \(x\) modulo \(N\) and an odd \(m\), write \((x/m)\) for the Jacobi symbol \((a/m)\) of the representative \(0 \le a < N\) of \(x\). Let \(U(m)\) be the group of units of \(\mathbb{Z}_m = \mathbb{Z}/m\mathbb{Z}\), let \(S(n') = \{y \in U(n') : (y/n') = 1\}\) and \(L(n;p') = \{x \in U(n) : (x/n) = (x/p')\}\), and for \(m \mid n\) let \(f_{n,m} : \mathbb{Z}_n \to \mathbb{Z}_m\) be reduction modulo \(m\). Then for every \(y \in S(n')\) and every \(z \in U(p')\) there is \(x \in L(n;p')\) with \(f_{n,n'}(x) = y\) and \(f_{n,p'}(x) = z\). That is, the Chinese remainder isomorphism \(\psi = (f_{n,n'}, f_{n,p'}) : U(n) \to U(n') \times U(p')\) satisfies \(S(n') \times U(p') \subseteq \psi(L(n;p'))\).

public theorem_wanted jacobi_kernel_prod_units_subset_crt_image (n p' : ℕ) (hn : Odd n)
    (hsf : Squarefree n) (hp' : p'.Prime) (hp'n : p' ∣ n) (y : ZMod (n / p'))
    (hy : y ∈ jacobiOneSet (n / p')) (z : ZMod p') (hz : IsUnit z) :
    ∃ x ∈ jacobiEqSet n p', ZMod.castHom (Nat.div_dvd_of_dvd hp'n) (ZMod (n / p')) x = y ∧
      ZMod.castHom hp'n (ZMod p') x = z

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Lemma 31 (jis-2024-24.1.7-lem-31).

TheoremWeighted zero-sum criterion for the Jacobi–Legendre weights#●ready to prove

Let \(n\) be a squarefree natural number all of whose prime divisors are at least \(7\), let \(p'\) be a prime divisor of \(n\), let \(n' = n/p'\), and let \(L(n;p')\) be the set of units \(x\) of \(\mathbb{Z}/n\mathbb{Z}\) whose Jacobi symbol \(\left(\frac{x}{n}\right)\) equals the Legendre symbol \(\left(\frac{x}{p'}\right)\). Let \((x_1, \dots, x_k)\) be a sequence in \(\mathbb{Z}/n\mathbb{Z}\) such that for every prime divisor \(p\) of \(n\) at least two terms \(x_i\) are not divisible by \(p\). Suppose that at most one term \(x_i\) maps to a unit under the reduction map \(\mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n'\mathbb{Z}\), or that for some prime divisor \(p\) of \(n'\) at least three terms \(x_i\) are not divisible by \(p\). Then there are weights \(a_1, \dots, a_k \in L(n;p')\) with \(a_1 x_1 + \dots + a_k x_k = 0\).

public theorem_wanted jacobi_eq_legendre_weighted_zero_sum (n p' k : ℕ) (hsf : Squarefree n)
    (h7 : ∀ p ∈ n.primeFactors, 7 ≤ p) (hp' : p' ∈ n.primeFactors) (x : Fin k → ZMod n)
    (h2 : ∀ p ∈ n.primeFactors, 2 ≤ (Finset.univ.filter fun i => Nat.Coprime (x i).val p).card)
    (hcase : (Finset.univ.filter fun i =>
        IsUnit (ZMod.castHom (Nat.div_dvd_of_dvd (Nat.dvd_of_mem_primeFactors hp'))
          (ZMod (n / p')) (x i))).card ≤ 1 ∨
      ∃ p ∈ (n / p').primeFactors,
        3 ≤ (Finset.univ.filter fun i => Nat.Coprime (x i).val p).card) :
    IsWeightedZeroSum n k (jacobiEqSet n p') x

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Lemma 32 (jis-2024-24.1.7-lem-32).

TheoremDavenport constant of the Jacobi–Legendre weights#●ready to prove

For a natural number \(n\) and \(A \subseteq \mathbb{Z}/n\mathbb{Z}\), let \(D_A(n)\) denote the least positive integer \(k\) such that every sequence \((x_1, \dots, x_k)\) in \(\mathbb{Z}/n\mathbb{Z}\) has a nonempty set \(J\) of indices and weights \(a_i \in A\) for \(i \in J\) with \(\sum_{i \in J} a_i x_i = 0\). Let \(n\) be a squarefree natural number all of whose prime divisors are at least \(7\), with \(\Omega(n) \ne 2\), where \(\Omega(n)\) is the number of prime factors of \(n\) counted with multiplicity. Let \(p'\) be a prime divisor of \(n\) and let \(L(n;p')\) be the set of units \(x\) of \(\mathbb{Z}/n\mathbb{Z}\) whose Jacobi symbol \(\left(\frac{x}{n}\right)\) equals the Legendre symbol \(\left(\frac{x}{p'}\right)\). Then \(D_{L(n;p')}(n) = \Omega(n) + 1\).

public theorem_wanted jacobi_eq_legendre_davenport_constant (n p' : ℕ) (hsf : Squarefree n)
    (h7 : ∀ p ∈ n.primeFactors, 7 ≤ p) (hΩ : Ω n ≠ 2) (hp' : p' ∈ n.primeFactors) :
    IsLeast (davenportSet n (jacobiEqSet n p')) (Ω n + 1)

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Theorem 33 (jis-2024-24.1.7-thm-33).

TheoremDavenport constant of the Jacobi–Legendre weights modulo a product of two primes#●ready to prove

For a natural number \(n\) and \(A \subseteq \mathbb{Z}/n\mathbb{Z}\), let \(D_A(n)\) denote the least positive integer \(k\) such that every sequence \((x_1, \dots, x_k)\) in \(\mathbb{Z}/n\mathbb{Z}\) has a nonempty set \(J\) of indices and weights \(a_i \in A\) for \(i \in J\) with \(\sum_{i \in J} a_i x_i = 0\). Let \(p'\) and \(q\) be distinct primes with \(p' \ge 7\) and \(q \ge 7\), let \(n = p'q\), and let \(L(n;p')\) be the set of units \(x\) of \(\mathbb{Z}/n\mathbb{Z}\) whose Jacobi symbol \(\left(\frac{x}{n}\right)\) equals the Legendre symbol \(\left(\frac{x}{p'}\right)\). Then \(D_{L(n;p')}(n) = 4\).

public theorem_wanted jacobi_eq_legendre_davenport_constant_two_primes (p' q : ℕ)
    (hp' : p'.Prime) (hq : q.Prime) (hne : p' ≠ q) (h7p : 7 ≤ p') (h7q : 7 ≤ q) :
    IsLeast (davenportSet (p' * q) (jacobiEqSet (p' * q) p')) 4

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Theorem 34 (jis-2024-24.1.7-thm-34).

TheoremConsecutive constant of the Jacobi–Legendre weights#●ready to prove

For a natural number \(n\) and \(A \subseteq \mathbb{Z}/n\mathbb{Z}\), let \(C_A(n)\) denote the least positive integer \(k\) such that every sequence \((x_1, \dots, x_k)\) in \(\mathbb{Z}/n\mathbb{Z}\) has a block of consecutive terms \(x_s, \dots, x_t\) with \(s \le t\) and weights \(a_s, \dots, a_t \in A\) such that \(a_s x_s + \dots + a_t x_t = 0\). Let \(n\) be a squarefree natural number all of whose prime divisors are at least \(7\), with \(\Omega(n) \ne 2\), where \(\Omega(n)\) is the number of prime factors of \(n\) counted with multiplicity. Let \(p'\) be a prime divisor of \(n\) and let \(L(n;p')\) be the set of units \(x\) of \(\mathbb{Z}/n\mathbb{Z}\) whose Jacobi symbol \(\left(\frac{x}{n}\right)\) equals the Legendre symbol \(\left(\frac{x}{p'}\right)\). Then \(C_{L(n;p')}(n) = 2^{\Omega(n)}\).

public theorem_wanted jacobi_eq_legendre_consecutive_constant (n p' : ℕ) (hsf : Squarefree n)
    (h7 : ∀ p ∈ n.primeFactors, 7 ≤ p) (hΩ : Ω n ≠ 2) (hp' : p' ∈ n.primeFactors) :
    IsLeast (consecutiveSet n (jacobiEqSet n p')) (2 ^ Ω n)

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Theorem 35 (jis-2024-24.1.7-thm-35).

TheoremConsecutive constant of the Jacobi–Legendre weights modulo a product of two primes#●ready to prove

For a natural number \(n\) and \(A \subseteq \mathbb{Z}/n\mathbb{Z}\), let \(C_A(n)\) denote the least positive integer \(k\) such that every sequence \((x_1, \dots, x_k)\) in \(\mathbb{Z}/n\mathbb{Z}\) has a block of consecutive terms \(x_s, \dots, x_t\) with \(s \le t\) and weights \(a_s, \dots, a_t \in A\) such that \(a_s x_s + \dots + a_t x_t = 0\). Let \(p'\) and \(q\) be distinct primes with \(p' \ge 7\) and \(q \ge 7\), let \(n = p'q\), and let \(L(n;p')\) be the set of units \(x\) of \(\mathbb{Z}/n\mathbb{Z}\) whose Jacobi symbol \(\left(\frac{x}{n}\right)\) equals the Legendre symbol \(\left(\frac{x}{p'}\right)\). Then \(C_{L(n;p')}(n) = 6\).

public theorem_wanted jacobi_eq_legendre_consecutive_constant_two_primes (p' q : ℕ)
    (hp' : p'.Prime) (hq : q.Prime) (hne : p' ≠ q) (h7p : 7 ≤ p') (h7q : 7 ≤ q) :
    IsLeast (consecutiveSet (p' * q) (jacobiEqSet (p' * q) p')) 6

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, Zero-Sum Constants Related to the Jacobi Symbol, Journal of Integer Sequences 27 (2024), Article 24.1.7, Theorem 36 (jis-2024-24.1.7-thm-36).

TheoremWeighted Davenport constant of a product with product weights#●ready to prove

Let \(R\) and \(R'\) be rings, \(M\) an \(R\)-module, \(N\) an \(R'\)-module, \(A \subseteq R\) and \(B \subseteq R'\). Let \(D_A(M)\) be the least positive integer \(k\), if one exists, such that every sequence \(x_1, \dots, x_k\) in \(M\) has a nonempty set of indices \(I\) and weights \(a_i \in A\) for \(i \in I\) with \(\sum_{i \in I} a_i x_i = 0\), and define \(D_B(N)\) in the same way. Let \(D_{A \times B}(M \times N)\) be the least positive integer \(k\), if one exists, such that every sequence \((y_1, z_1), \dots, (y_k, z_k)\) in \(M \times N\) has a nonempty set of indices \(I\) and weights \((a_i, b_i) \in A \times B\) for \(i \in I\) with \(\sum_{i \in I} (a_i y_i, b_i z_i) = 0\). If all three constants exist, then \(D_{A \times B}(M \times N) \ge D_A(M) + D_B(N) - 1\).

public theorem_wanted davenport_prod_ring_ge {R R' M N : Type*} [Ring R] [Ring R']
    [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R' N] (A : Set R) (B : Set R')
    {dA dB d : ℕ} (hA : IsLeast (davenportSet R M A) dA)
    (hB : IsLeast (davenportSet R' N B) dB)
    (h : IsLeast {k : ℕ | 0 < k ∧ ∀ x : Fin k → M × N, ∃ I : Finset (Fin k), I.Nonempty ∧
      ∃ a : Fin k → R × R', (∀ i ∈ I, a i ∈ A ×ˢ B) ∧
        ∑ i ∈ I, ((a i).1 • (x i).1, (a i).2 • (x i).2) = 0} d) :
    dA + dB - 1 ≤ d

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, On Unit-Weighted Zero-Sum Constants of ℤ_n, INTEGERS 24 (2024), Article A38, Theorem 2.1.

TheoremWeighted Davenport constant of a product of modules#●ready to prove

Let \(R\) be a ring and \(A \subseteq R\). For an \(R\)-module \(P\), let \(D_A(P)\) be the least positive integer \(k\), if one exists, such that every sequence \(x_1, \dots, x_k\) in \(P\) has a nonempty set of indices \(I\) and weights \(a_i \in A\) for \(i \in I\) with \(\sum_{i \in I} a_i x_i = 0\). If \(M\) and \(N\) are \(R\)-modules for which \(D_A(M)\), \(D_A(N)\) and \(D_A(M \times N)\) all exist, then \(D_A(M \times N) \ge D_A(M) + D_A(N) - 1\).

public theorem_wanted davenport_prod_ge {R M N : Type*} [Ring R] [AddCommGroup M] [Module R M]
    [AddCommGroup N] [Module R N] (A : Set R) {dM dN d : ℕ}
    (hM : IsLeast (davenportSet R M A) dM) (hN : IsLeast (davenportSet R N A) dN)
    (h : IsLeast (davenportSet R (M × N) A) d) :
    dM + dN - 1 ≤ d

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, On Unit-Weighted Zero-Sum Constants of ℤ_n, INTEGERS 24 (2024), Article A38, Theorem 2.2.

TheoremConsecutive weighted zero-sum constant of a product with product weights#●ready to prove

Let \(R\) and \(R'\) be rings, \(M\) an \(R\)-module, \(N\) an \(R'\)-module, \(A \subseteq R\) and \(B \subseteq R'\). Let \(C_A(M)\) be the least positive integer \(k\), if one exists, such that every sequence \(x_1, \dots, x_k\) in \(M\) has indices \(s \le t\) and weights \(a_s, \dots, a_t \in A\) with \(a_s x_s + \dots + a_t x_t = 0\), and define \(C_B(N)\) in the same way. Let \(C_{A \times B}(M \times N)\) be the least positive integer \(k\), if one exists, such that every sequence \((y_1, z_1), \dots, (y_k, z_k)\) in \(M \times N\) has indices \(s \le t\) and weights \((a_i, b_i) \in A \times B\) for \(s \le i \le t\) with \(\sum_{i=s}^{t} (a_i y_i, b_i z_i) = 0\). If all three constants exist, then \(C_{A \times B}(M \times N) \ge C_A(M) \, C_B(N)\).

public theorem_wanted consecutive_prod_ring_ge {R R' M N : Type*} [Ring R] [Ring R']
    [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R' N] (A : Set R) (B : Set R')
    {cA cB c : ℕ} (hA : IsLeast (consecutiveSet R M A) cA)
    (hB : IsLeast (consecutiveSet R' N B) cB)
    (h : IsLeast {k : ℕ | 0 < k ∧ ∀ x : Fin k → M × N, ∃ s t : Fin k, s ≤ t ∧
      ∃ a : Fin k → R × R', (∀ i ∈ Finset.Icc s t, a i ∈ A ×ˢ B) ∧
        ∑ i ∈ Finset.Icc s t, ((a i).1 • (x i).1, (a i).2 • (x i).2) = 0} c) :
    cA * cB ≤ c

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, On Unit-Weighted Zero-Sum Constants of ℤ_n, INTEGERS 24 (2024), Article A38, Theorem 2.3.

TheoremConsecutive weighted zero-sum constant of a product of modules#●ready to prove

Let \(R\) be a ring and \(A \subseteq R\). For an \(R\)-module \(P\), let \(C_A(P)\) be the least positive integer \(k\), if one exists, such that every sequence \(x_1, \dots, x_k\) in \(P\) has indices \(s \le t\) and weights \(a_s, \dots, a_t \in A\) with \(a_s x_s + \dots + a_t x_t = 0\). If \(M\) and \(N\) are \(R\)-modules for which \(C_A(M)\), \(C_A(N)\) and \(C_A(M \times N)\) all exist, then \(C_A(M \times N) \ge C_A(M) \, C_A(N)\).

public theorem_wanted consecutive_prod_ge {R M N : Type*} [Ring R] [AddCommGroup M]
    [Module R M] [AddCommGroup N] [Module R N] (A : Set R) {cM cN c : ℕ}
    (hM : IsLeast (consecutiveSet R M A) cM) (hN : IsLeast (consecutiveSet R N A) cN)
    (h : IsLeast (consecutiveSet R (M × N) A) c) :
    cM * cN ≤ c

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, On Unit-Weighted Zero-Sum Constants of ℤ_n, INTEGERS 24 (2024), Article A38, Theorem 2.4.

TheoremConsecutive zero-sum constant of a finite abelian group#●ready to prove

Let \(G\) be a finite abelian group and let \(C(G)\) be the least positive integer \(k\) such that every sequence \(x_1, \dots, x_k\) in \(G\) has a block of consecutive terms \(x_s, \dots, x_t\) with \(s \le t\) and \(x_s + \dots + x_t = 0\). Then \(C(G) = |G|\).

public theorem_wanted consecutive_constant_eq_card (G : Type*) [AddCommGroup G] [Fintype G] :
    IsLeast (consecutiveSet ℤ G ({1} : Set ℤ)) (Fintype.card G)

Sources

• Santanu Mondal, Krishnendu Paul and Shameek Paul, On Unit-Weighted Zero-Sum Constants of ℤ_n, INTEGERS 24 (2024), Article A38, Theorem 3.1.

20 ready to prove

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