Displaying similar documents to “On the lattice of congruences on inverse semirings”

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...

Diamond identities for relative congruences

Gábor Czédli (1995)

Archivum Mathematicum

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For a class K of structures and A K let C o n * ( A ) resp. C o n K ( A ) denote the lattices of * -congruences resp. K -congruences of A , cf. Weaver [25]. Let C o n * ( K ) : = I { C o n * ( A ) : A K } where I is the operator of forming isomorphic copies, and C o n r ( K ) : = I { C o n K ( A ) : A K } . For an ordered algebra A the lattice of order congruences of A is denoted by C o n < ( A ) , and let C o n < ( K ) : = I { C o n < ( A ) : A K } if K is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by Q s and P , respectively. Let λ be a lattice identity and let Σ be a set of lattice identities....

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.

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 ] .

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 ) .

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 2 p - 1 p - 1 1 ( mod p 4 ) . For such a prime p , we establish an expression for 2 p - 1 p - 1 ( mod p 8 ) given in terms of the sums R i : = 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 2 p - 1 p - 1 1 - 2 p k = 1 p - 1 1 k - 2 p 2 k = 1 p - 1 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...

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 ) 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 ] ) g ( [ 0 , M ] ) | M 1 - ε .

Non-abelian p -adic L -functions and Eisenstein series of unitary groups – The CM method

Thanasis Bouganis (2014)

Annales de l’institut Fourier

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In this work we prove various cases of the so-called “torsion congruences” between abelian p -adic L -functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of n variables and we obtain more explicit results...

On a congruence of Emma Lehmer related to Euler numbers

John B. Cosgrave, Karl Dilcher (2013)

Acta Arithmetica

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A congruence of Emma Lehmer (1938) for Euler numbers E p - 3 modulo p in terms of a certain sum of reciprocals of squares of integers was recently extended to prime power moduli by T. Cai et al. We generalize this further to arbitrary composite moduli n and characterize those n for which the sum in question vanishes modulo n (or modulo n/3 when 3|n). Primes for which E p - 3 0 ( m o d p ) play an important role, and we present some numerical results.