Principal congruence link complements

Mark D. Baker; Alan W. Reid

Displaying similar documents to “Principal congruence link complements”

The existence of solutions of the generalized pseudoprime congruence $a^{f (n)} \equiv b^{f (n)} (m o d n)$

Wayne L. McDaniel (1990)

Colloquium Mathematicae

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Congruence preserving operations on the ring $ℤ_{p^{3}}$

Cyril Gavala, Miroslav Ploščica, Ivana Varga (2023)

Mathematica Bohemica

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We investigate the interval $I (p^{3})$ in the lattice of clones on the ring $ℤ_{p^{3}}$ between the clone of polynomial operations and the clone of congruence preserving operations. All clones in this interval are known and described by means of generators. In this paper, we characterize each of these clones by the property of preserving a small set of relations. These relations turn out to be in a close connection to commutators.

On the lattice of congruences on inverse semirings

Anwesha Bhuniya, Anjan Kumar Bhuniya (2008)

Discussiones Mathematicae - General Algebra and Applications

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Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences $ρ_{m i n}, ρ_{m a x}, ρ^{m i n}$ and $ρ^{m a x}$ on S and showed that $ρ θ = [ρ_{m i n}, ρ_{m a x}]$ and $ρ κ = [ρ^{m i n}, ρ^{m a x}]$ . Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if $ρ_{m a x}$ is a distributive lattice congruence and $ρ^{m a x}$ is a skew-ring congruence on S. If η (σ) is the...

On a linear homogeneous congruence

A. Schinzel, M. Zakarczemny (2006)

Colloquium Mathematicae

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The number of solutions of the congruence $a ₁ x ₁ + \dots + a_{k} x_{k} \equiv 0 (m o d n)$ in the box $0 \leq x_{i} \leq b_{i}$ is estimated from below in the best possible way, provided for all i,j either $(a_{i}, n) | (a_{j}, n)$ or $(a_{j}, n) | (a_{i}, n)$ or $n | [a_{i}, a_{j}]$ .

Congruences for Wolstenholme primes

Romeo Meštrović (2015)

Czechoslovak Mathematical Journal

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A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence $(\binom{2 p - 1}{p - 1}) \equiv 1 (mod p^{4})$ . For such a prime $p$ , we establish an expression for $(\binom{2 p - 1}{p - 1}) (mod p^{8})$ given in terms of the sums $R_{i} : = \sum_{k = 1}^{p - 1} 1 / k^{i}$ ( $i = 1, 2, 3, 4, 5, 6)$ . Further, the expression in this congruence is reduced in terms of the sums $R_{i}$ ( $i = 1, 3, 4, 5$ ). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $(\binom{2 p - 1}{p - 1}) \equiv 1 - 2 p \sum_{k = 1}^{p - 1} \frac{1}{k} - 2 p^{2} \sum_{k = 1}^{p - 1} \frac{1}{k^{2}} (mod p^{7}) .$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying...

On the arithmetic of arithmetical congruence monoids

M. Banister, J. Chaika, S. T. Chapman, W. Meyerson (2007)

Colloquium Mathematicae

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Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of $ℤ_{b} = ℤ / b ℤ$ , then the set $H_{Γ} = x \in ℕ | x + b ℤ \in Γ \cup 1$ is a multiplicative submonoid of ℕ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = ā consists of a single element. If $H_{Γ}$ is an ACM, then we represent it with the notation M(a,b) = (a + bℕ₀) ∪ 1, where a, b ∈ ℕ and a² ≡ a (mod b). A classical 1954 result of James and Niven implies that the...

On congruence permutable $G$ -sets

Attila Nagy (2020)

Commentationes Mathematicae Universitatis Carolinae

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An algebraic structure is said to be congruence permutable if its arbitrary congruences $α$ and $β$ satisfy the equation $α \circ β = β \circ α$ , where $\circ$ denotes the usual composition of binary relations. To an arbitrary $G$ -set $X$ satisfying $G \cap X = \emptyset$ , we assign a semigroup $(G, X, 0)$ on the base set $G \cup X \cup {0}$ containing a zero element $0 \notin G \cup X$ , and examine the connection between the congruence permutability of the $G$ -set $X$ and the semigroup $(G, X, 0)$ .

Some monounary algebras with EKP

Emília Halušková (2020)

Mathematica Bohemica

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An algebra $𝒜$ is said to have the endomorphism kernel property (EKP) if every congruence on $𝒜$ is the kernel of some endomorphism of $𝒜$ . Three classes of monounary algebras are dealt with. For these classes, all monounary algebras with EKP are described.

Linear congruences and a conjecture of Bibak

Chinnakonda Gnanamoorthy Karthick Babu, Ranjan Bera, Balasubramanian Sury (2024)

Czechoslovak Mathematical Journal

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We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences $\sum_{i = 1}^{k} a_{i} x_{i} \equiv b (mod n)$ . In particular, we obtain explicit expressions for the number of solutions, where $x_{i}$ ’s are squares modulo $n$ . In addition, we obtain expressions for the number of solutions with order restrictions $x_{1} \geq \dots \geq x_{k}$ or with strict order restrictions $x_{1} > \dots > x_{k}$ in some special cases. In these results, the expressions for the number of solutions involve...

Quantitative spectral gap for thin groups of hyperbolic isometries

Michael Magee (2015)

Journal of the European Mathematical Society

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Let $Λ$ be a subgroup of an arithmetic lattice in $SO (n + 1, 1)$ . The quotient $ℍ^{n + 1} / Λ$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $Λ$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Zhi-Hong Sun (2011)

Acta Arithmetica

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A $q$ -congruence for a truncated $_{4} ϕ_{3}$ series

Victor J. W. Guo, Chuanan Wei (2021)

Czechoslovak Mathematical Journal

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Let $Φ_{n} (q)$ denote the $n$ th cyclotomic polynomial in $q$ . Recently, Guo, Schlosser and Zudilin proved that for any integer $n > 1$ with $n \equiv 1 (mod 4)$ , $\sum_{k = 0}^{n - 1} \frac{{(q^{- 1}; q^{2})}_{k}^{2} {(q^{- 2}; q^{4})}_{k}}{{(q^{2}; q^{2})}_{k}^{2} {(q^{4}; q^{4})}_{k}} q^{6 k} \equiv 0 (mod Φ_{n} {(q)}^{2}),$ where ${(a; q)}_{m} = (1 - a) (1 - a q) \dots (1 - a q^{m - 1})$ . In this note, we give a generalization of the above $q$ -congruence to the modulus $Φ_{n} {(q)}^{3}$ case. Meanwhile, we give a corresponding $q$ -congruence modulo $Φ_{n} {(q)}^{2}$ for $n \equiv 3 (mod 4)$ . Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a $_{4} ϕ_{3}$ summation formula.

Endomorphism kernel property for finite groups

Heghine Ghumashyan, Jaroslav Guričan (2022)

Mathematica Bohemica

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A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $θ$ on $G$ is the kernel of an endomorphism on $G$ . In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.

Sparsity of the intersection of polynomial images of an interval

Mei-Chu Chang (2014)

Acta Arithmetica

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We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let $f (x), g (x) \in_{p} [x]$ be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies $p^{1 / E (1 + κ / (1 - κ)} > M > p^{ε}$ , where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then $| f ([0, M]) \cap g ([0, M]) | ≲ M^{1 - ε}$ .

Some properties of the class of arithmetic functions $T_{r} (N)$

R. P. Pakshirajan (1963)

Annales Polonici Mathematici

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A formula for the number of solutions of a restricted linear congruence

K. Vishnu Namboothiri (2021)

Mathematica Bohemica

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Consider the linear congruence equation $x_{1} + ... + x_{k} \equiv b (mod n^{s})$ for $b \in ℤ$ , $n, s \in ℕ$ . Let ${(a, b)}_{s}$ denote the generalized gcd of $a$ and $b$ which is the largest $l^{s}$ with $l \in ℕ$ dividing $a$ and $b$ simultaneously. Let $d_{1}, ..., d_{τ (n)}$ be all positive divisors of $n$ . For each $d_{j} ∣ n$ , define $𝒞_{j, s} (n) = {1 \leq x \leq n^{s} : {(x, n^{s})}_{s} = d_{j}^{s}}$ . K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_{i}$ . We generalize their result with generalized gcd restrictions on $x_{i}$ and prove that for the above linear congruence, the...

Witten's Conjecture for many four-manifolds of simple type

Paul M. N. Feehan, Thomas G. Leness (2015)

Journal of the European Mathematical Society

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We prove that Witten’s Conjecture [40] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with $b_{1} = 0$ and odd $b_{2}^{+} \geq 3$ follows from our $(3)$ -monopole cobordism formula [6] when the four-manifold has $c_{1}^{2} \geq χ_{h} - 3$ or is abundant.