The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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.