Displaying similar documents to “On sums of binomial coefficients modulo p²”

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 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.

Some q-supercongruences for truncated basic hypergeometric series

Victor J. W. Guo, Jiang Zeng (2015)

Acta Arithmetica

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For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as k = 0 ( p - 1 ) / 2 [ 2 k k ] q ² 3 ( q 2 k ) / ( ( - q ² ; q ² ) ² k ( - q ; q ) ² 2 k ² ) 0 ( m o d [ p ] ² ) for p≡ 3 (mod 4), k = 0 ( p - 1 ) / 2 [ 2 k k ] q ³ ( ( q ; q ³ ) k ( q ² ; q ³ ) k q 3 k ) ( ( q ; q ) k ² ) 0 ( m o d [ p ] ² ) for p≡ 2 (mod 3), where [ p ] = 1 + q + + q p - 1 and ( a ; q ) = ( 1 - a ) ( 1 - a q ) ( 1 - a q n - 1 ) . We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula. ...

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.

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...

Linear congruences and a conjecture of Bibak

Chinnakonda Gnanamoorthy Karthick Babu, Ranjan Bera, Balasubramanian Sury (2024)

Czechoslovak Mathematical Journal

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We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences i = 1 k a i x i b ( mod n ) . In particular, we obtain explicit expressions for the number of solutions, where x i ’s are squares modulo n . In addition, we obtain expressions for the number of solutions with order restrictions x 1 x k or with strict order restrictions x 1 > > x k in some special cases. In these results, the expressions for the number of solutions involve...

On a family of elliptic curves of rank at least 2

Kalyan Chakraborty, Richa Sharma (2022)

Czechoslovak Mathematical Journal

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Let C m : y 2 = x 3 - m 2 x + p 2 q 2 be a family of elliptic curves over , where m is a positive integer and p , q are distinct odd primes. We study the torsion part and the rank of C m ( ) . More specifically, we prove that the torsion subgroup of C m ( ) is trivial and the -rank of this family is at least 2, whenever m ¬ 0 ( mod 3 ) , m ¬ 0 ( mod 4 ) and m 2 ( mod 64 ) with neither p nor q dividing m .

Integral points on the elliptic curve y 2 = x 3 - 4 p 2 x

Hai Yang, Ruiqin Fu (2019)

Czechoslovak Mathematical Journal

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Let p be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve E : y 2 = x 3 - 4 p 2 x . Further, let N ( p ) denote the number of pairs of integral points ( x , ± y ) on E with y > 0 . We prove that if p 17 , then N ( p ) 4 or 1 depending on whether p 1 ( mod 8 ) or p - 1 ( mod 8 ) .

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, ...

Some new sums related to D. H. Lehmer problem

Han Zhang, Wenpeng Zhang (2015)

Czechoslovak Mathematical Journal

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About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N ( k ; p ) denote the number of all 1 a i p - 1 such that a 1 a 2 a k 1 mod p and 2 a i + a ¯ i + 1 , i = 1 , 2 , , k . The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N ( k ; p ) , and give an interesting asymptotic formula...

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...

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...