Displaying similar documents to “Congruences for ( A + ( A ² + m B ² ) ) ( p - 1 ) / 2 ) and ( b + ( a ² + b ² ) ) ( p - 1 ) / 4 ( m o d p )

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...

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.

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 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 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...

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 C * -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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