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Displaying similar documents to “A q -congruence for a truncated 4 ϕ 3 series”

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 b ( mod n s ) for b , n , s . Let ( a , b ) s denote the generalized gcd of a and b which is the largest l s with l 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 x 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...

On the Lucas sequence equations Vₙ = kVₘ and Uₙ = kUₘ

Refik Keskin, Zafer Şiar (2013)

Colloquium Mathematicae

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Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and U n + 1 = P U - Q U n - 1 for n ≥ 1, and V₀ = 2, V₁ = P and V n + 1 = P V - Q V n - 1 for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and V r 1 . We show that there is no integer x such that V = V r V x ² when m ≥ 1 and r is an even integer. Also we completely solve the equation V = V V r x ² for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...

Polynomials, sign patterns and Descartes' rule of signs

Vladimir Petrov Kostov (2019)

Mathematica Bohemica

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By Descartes’ rule of signs, a real degree d polynomial P with all nonvanishing coefficients with c sign changes and p sign preservations in the sequence of its coefficients ( c + p = d ) has pos c positive and ¬ p negative roots, where pos c ( mod 2 ) and ¬ p ( mod 2 ) . For 1 d 3 , for every possible choice of the sequence of signs of coefficients of P (called sign pattern) and for every pair ( pos , neg ) satisfying these conditions there exists a polynomial P with exactly pos positive and exactly ¬ negative roots (all of them simple). For d 4 ...

On power integral bases for certain pure number fields defined by x 18 - m

Lhoussain El Fadil (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let K = ( α ) be a number field generated by a complex root α of a monic irreducible polynomial f ( x ) = x 18 - m , m 1 , is a square free rational integer. We prove that if m 2 or 3 ( mod 4 ) and m ¬ 1 ( mod 9 ) , then the number field K is monogenic. If m 1 ( mod 4 ) or m 1 ( mod 9 ) , then the number field K is not monogenic.

Modular symbols, Eisenstein series, and congruences

Jay Heumann, Vinayak Vatsal (2014)

Journal de Théorie des Nombres de Bordeaux

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Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k 2 and of the same level N , both eigenfunctions of the Hecke operators, and both normalized so that a 1 ( f ) = a 1 ( E ) = 1 . The main result we prove is that when E and f are congruent mod a prime 𝔭 (which we take in this paper to be a prime of ¯ lying over a rational prime p > 2 ), the algebraic parts of the special values L ( E , χ , j ) and L ( f , χ , j ) satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...

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 β = β α , where denotes the usual composition of binary relations. To an arbitrary G -set X satisfying G X = , we assign a semigroup ( G , X , 0 ) on the base set G X { 0 } containing a zero element 0 G X , and examine the connection between the congruence permutability of the G -set X and the semigroup ( G , X , 0 ) .

Automorphisms of metacyclic groups

Haimiao Chen, Yueshan Xiong, Zhongjian Zhu (2018)

Czechoslovak Mathematical Journal

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A metacyclic group H can be presented as α , β : α n = 1 , β m = α t , β α β - 1 = α r for some n , m , t , r . Each endomorphism σ of H is determined by σ ( α ) = α x 1 β y 1 , σ ( β ) = α x 2 β y 2 for some integers x 1 , x 2 , y 1 , y 2 . We give sufficient and necessary conditions on x 1 , x 2 , y 1 , y 2 for σ to be an automorphism.

The Rothberger property on C p ( Ψ ( 𝒜 ) , 2 )

Daniel Bernal-Santos (2016)

Commentationes Mathematicae Universitatis Carolinae

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A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 𝒰 n : n ω of open covers of X , there exists U n 𝒰 n for each n ω such that X = n ω U n . For any n ω , necessary and sufficient conditions are obtained for C p ( Ψ ( 𝒜 ) , 2 ) n to have the Rothberger property when 𝒜 is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family 𝒜 for which the space C p ( Ψ ( 𝒜 ) , 2 ) n is Rothberger for all n ω .

Involutivity degree of a distribution at superdensity points of its tangencies

Silvano Delladio (2021)

Archivum Mathematicum

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Let Φ 1 , ... , Φ k + 1 (with k 1 ) be vector fields of class C k in an open set U N + m , let 𝕄 be a N -dimensional C k submanifold of U and define 𝕋 : = { z 𝕄 : Φ 1 ( z ) , ... , Φ k + 1 ( z ) T z 𝕄 } where T z 𝕄 is the tangent space to 𝕄 at z . Then we expect the following property, which is obvious in the special case when z 0 is an interior point (relative to 𝕄 ) of 𝕋 : If z 0 𝕄 is a ( N + k ) -density point (relative to 𝕄 ) of 𝕋 then all the iterated Lie brackets of order less or equal to k Φ i 1 ( z 0 ) , [ Φ i 1 , Φ i 2 ] ( z 0 ) , [ [ Φ i 1 , Φ i 2 ] , Φ i 3 ] ( z 0 ) , ... ( h , i h k + 1 ) belong to T z 0 𝕄 . Such a property has been proved in [9] for k = 1 and its proof in the...

On the exponential diophantine equation x y + y x = z z

Xiaoying Du (2017)

Czechoslovak Mathematical Journal

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For any positive integer D which is not a square, let ( u 1 , v 1 ) be the least positive integer solution of the Pell equation u 2 - D v 2 = 1 , and let h ( 4 D ) denote the class number of binary quadratic primitive forms of discriminant 4 D . If D satisfies 2 D and v 1 h ( 4 D ) 0 ( mod D ) , then D is called a singular number. In this paper, we prove that if ( x , y , z ) is a positive integer solution of the equation x y + y x = z z with 2 z , then maximum max { x , y , z } < 480000 and both x , y are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions...

A localization property for B p q s and F p q s spaces

Hans Triebel (1994)

Studia Mathematica

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Let f j = k a k f ( 2 j + 1 x - 2 k ) , where the sum is taken over the lattice of all points k in n having integer-valued components, j∈ℕ and a k . Let A p q s be either B p q s or F p q s (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on n . The aim of the paper is to clarify under what conditions f j | A p q s is equivalent to 2 j ( s - n / p ) ( k | a k | p ) 1 / p f | A p q s .

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.