Displaying similar documents to “Linear congruences and a conjecture of Bibak”

New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs

Ernest X. W. Xia (2015)

Colloquium Mathematicae

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Let p p ¯ ( n ) denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, p p ¯ ( 3 n + 2 ) 0 ( m o d 3 ) . They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for p p ¯ ( n ) . Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for p p ¯ ( n ) . Furthermore, they also constructed infinite families of congruences for p p ¯ ( n ) modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several...

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 1 ( mod 4 ) , k = 0 n - 1 ( 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 0 ( mod Φ n ( q ) 2 ) , where ( a ; q ) m = ( 1 - a ) ( 1 - a q ) ( 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 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.

Congruences and homomorphisms on Ω -algebras

Elijah Eghosa Edeghagba, Branimir Šešelja, Andreja Tepavčević (2017)

Kybernetika

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The topic of the paper are Ω -algebras, where Ω is a complete lattice. In this research we deal with congruences and homomorphisms. An Ω -algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an Ω -valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce Ω -valued congruences, corresponding quotient Ω -algebras and Ω -homomorphisms and we investigate connections among these notions....

On Alternatives of Polynomial Congruences

Mariusz Skałba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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What should be assumed about the integral polynomials f ( x ) , . . . , f k ( x ) in order that the solvability of the congruence f ( x ) f ( x ) f k ( x ) 0 ( m o d p ) for sufficiently large primes p implies the solvability of the equation f ( x ) f ( x ) f k ( x ) = 0 in integers x? We provide some explicit characterizations for the cases when f j ( x ) are binomials or have cyclic splitting fields.

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 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 ) .

Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Ernest X. W. Xia (2016)

Colloquium Mathematicae

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Let p ̅ o ( n ) denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function p ̅ o ( n ) have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for p ̅ o ( n ) modulo 3. For example, we prove that for n, α ≥ 0, p ̅ o ( 4 α ( 24 n + 17 ) ) p ̅ o ( 4 α ( 24 n + 23 ) ) 0 ( m o d 3 ) .

On a linear homogeneous congruence

A. Schinzel, M. Zakarczemny (2006)

Colloquium Mathematicae

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The number of solutions of the congruence a x + + a k x k 0 ( m o d n ) in the box 0 x i 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 ] .

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...

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 the quartic character of quadratic units

Zhi-Hong Sun (2013)

Acta Arithmetica

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Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of integers m and n. Let p be a prime of the form 4k+1 and p = c²+d² with c,d ∈ ℤ, d = 2 r d and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine ( b + ( b ² + 4 α ) / 2 ) ( p - 1 ) / 4 ) ( m o d p ) for p = x²+(b²+4α)y² (b,x,y ∈ ℤ, 2∤b), and ( 2 a + 4 a ² + 1 ) ( p - 1 ) / 4 ( m o d p ) for p = x²+(4a²+1)y² (a,x,y∈ℤ) on the condition that (c,x+d) = 1 or (d₀,x+c) = 1. As applications we obtain the congruence for U ( p - 1 ) / 4 ( m o d p ) and the criterion for p | U ( p - 1 ) / 8 (if p ≡ 1 (mod 8)), where Uₙ is the Lucas sequence given by U₀ = 0, U₁ = 1 and...

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Soufiane Mezroui (2020)

Czechoslovak Mathematical Journal

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Let f = n = 1 a ( n ) q n S k + 1 / 2 ( N , χ 0 ) be a nonzero cuspidal Hecke eigenform of weight k + 1 2 and the trivial nebentypus χ 0 , where the Fourier coefficients a ( n ) are real. Bruinier and Kohnen conjectured that the signs of a ( n ) are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies { a ( t n 2 ) } n , where t is a squarefree integer such that a ( t ) 0 . Let q and d be natural numbers such that ( d , q ) = 1 . In this work, we show that { a ( t n 2 ) } n is equidistributed over any arithmetic progression n d mod q .

On sums of binomial coefficients modulo p²

Zhi-Wei Sun (2012)

Colloquium Mathematicae

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Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum k = 0 p a - 1 ( h p a - 1 k ) ( 2 k k ) / m k ( m o d p ² ) , where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and p a > 3 , then k = 0 p a - 1 ( h p a - 1 k ) ( 2 k k ) ( - h / 2 ) k ( ( 1 - 2 h ) / ( p a ) ) ( 1 + h ( ( 4 - 2 / h ) p - 1 - 1 ) ) ( m o d p ² ) , where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If p a > 3 then k = 0 p a - 1 ( p a - 1 k ) ( 2 k k ) ( - 1 ) k 3 p - 1 ( p a / 3 ) ( m o d p ² ) .

On a conjecture of Dekking : The sum of digits of even numbers

Iurie Boreico, Daniel El-Baz, Thomas Stoll (2014)

Journal de Théorie des Nombres de Bordeaux

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Let q 2 and denote by s q the sum-of-digits function in base q . For j = 0 , 1 , , q - 1 consider # { 0 n < N : s q ( 2 n ) j ( mod q ) } . In 1983, F. M. Dekking conjectured that this quantity is greater than N / q and, respectively, less than N / q for infinitely many N , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

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 ...

Invariance of the parity conjecture for p -Selmer groups of elliptic curves in a D 2 p n -extension

Thomas de La Rochefoucauld (2011)

Bulletin de la Société Mathématique de France

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We show a p -parity result in a D 2 p n -extension of number fields L / K ( p 5 ) for the twist 1 η τ : W ( E / K , 1 η τ ) = ( - 1 ) 1 η τ , X p ( E / L ) , where E is an elliptic curve over K , η and τ are respectively the quadratic character and an irreductible representation of degree 2 of Gal ( L / K ) = D 2 p n , and X p ( E / L ) is the p -Selmer group. The main novelty is that we use a congruence result between ε 0 -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the p -parity conjecture...

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.

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.