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26
POLYNOMIAL EXTENSION OF FLECK’S CONGRUENCE
, 2005
"... Let p be a prime, and let f(x) be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the p-adic order of the sum k≡r (mod p β) n ..."
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Cited by 18 (9 self)
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Let p be a prime, and let f(x) be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the p-adic order of the sum k≡r (mod p β) n
VARIOUS CONGRUENCES INVOLVING BINOMIAL COEFFICIENTS AND HIGHER-ORDER CATALAN NUMBERS
, 2009
"... Let p be a prime and let a be a positive integer. In this paper we investigate Pp a −1 (h+1)k ´ k=0 /mk modulo a prime p, where d and m are k+d integers with −h < d � pa and m ̸ ≡ 0 (mod p). We also study congruences involving higher-order Catalan numbers C (h) ` 1 (h+1)k ´ k = and ¯(h) C hk+1 ..."
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Cited by 9 (9 self)
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Let p be a prime and let a be a positive integer. In this paper we investigate Pp a −1 (h+1)k ´ k=0 /mk modulo a prime p, where d and m are k+d integers with −h < d � pa and m ̸ ≡ 0 (mod p). We also study congruences involving higher-order Catalan numbers C (h) ` 1 (h+1)k ´ k = and ¯(h) C hk+1 k k = h (h+1)k ´. Our tools include linear recurrences and the theory of cubic k+1 k residues. Here are some typical results in the paper. (i) If pa ≡ 1 (mod 6) then Also, p a −1 X k=0 (ii) We have p a −1 X k=0
SOME CONGRUENCES FOR THE SECOND-ORDER CATALAN NUMBERS
- PROC. AMER. MATH. SOC. 138(2010), NO. 1, 37–46.
, 2010
"... Let p be any odd prime. We mainly show that p−1 k=1 ..."
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Cited by 9 (8 self)
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Let p be any odd prime. We mainly show that p−1 k=1
COVERS OF THE INTEGERS WITH ODD MODULI AND THEIR APPLICATIONS TO THE FORMS x m − 2 n AND x 2 − F3n/2
"... Abstract. In this paper we construct a cover {as(mod ns)} k s=1 of Z with odd moduli such that there are distinct primes p1,...,pk dividing 2n1 −1,...,2nk − 1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmet ..."
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Cited by 8 (2 self)
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Abstract. In this paper we construct a cover {as(mod ns)} k s=1 of Z with odd moduli such that there are distinct primes p1,...,pk dividing 2n1 −1,...,2nk − 1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the mth powers of whose terms are never of the form 2n ± pa with a, n ∈{0, 1, 2,...} and p a prime. We also construct another cover of Z with odd moduli and use it to prove that x2 − F3n/2 has at least two distinct prime factors whenever n ∈{0, 1, 2,...} and x ≡ a (mod M), where {Fi}i�0 is the Fibonacci sequence, and a and M are suitable positive integers having 80 decimal digits. 1.
A GENERALIZATION OF WOLSTENHOLME’S HARMONIC SERIES CONGRUENCE
, 2006
"... Abstract. Let A, B be two non-zero integers. Define the Lucas sequences {un} ∞ n=0 and {vn} ∞ n=0 by and u0 = 0, u1 = 1, un = Aun−1 − Bun−2 for n ≥ 2 v0 = 2, v1 = A, vn = Avn−1 − Bvn−2 for n ≥ 2. For any n ∈ Z +, let wn be the largest divisor of un prime to u1, u2,..., un−1. We prove that for any ..."
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Cited by 4 (0 self)
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Abstract. Let A, B be two non-zero integers. Define the Lucas sequences {un} ∞ n=0 and {vn} ∞ n=0 by and u0 = 0, u1 = 1, un = Aun−1 − Bun−2 for n ≥ 2 v0 = 2, v1 = A, vn = Avn−1 − Bvn−2 for n ≥ 2. For any n ∈ Z +, let wn be the largest divisor of un prime to u1, u2,..., un−1. We prove that for any n ≥ 5 where ∆ = A 2 − 4B. n−1 vj uj j=1
ON FLECK QUOTIENTS
- ACTA ARITH. 127(2007), NO. 4, 337–363.
, 2007
"... Let p be a prime, and let n � 1 and r be integers. In this paper we study Fleck’s quotient Fp(n, r) = (−p) −⌊(n−1)/(p−1)⌋ k≡r (mod p) n ..."
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Cited by 4 (0 self)
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Let p be a prime, and let n � 1 and r be integers. In this paper we study Fleck’s quotient Fp(n, r) = (−p) −⌊(n−1)/(p−1)⌋ k≡r (mod p) n
LUCAS-TYPE CONGRUENCES FOR CYCLOTOMIC ψ-COEFFICIENTS
- INT. J. NUMBER THEORY 4(2008), NO. 2, 155–170.
, 2008
"... Let p be any prime and a be a positive integer. For l, n ∈ {0, 1,...} and r ∈ Z, the normalized cyclotomic ψ-coefficient n: = p− r l,pa ⌊ n−p a−1 −lp a p a−1 (p−1) k≡r (mod p a) (−1) k ( n) ( k−r p k ..."
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Cited by 3 (2 self)
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Let p be any prime and a be a positive integer. For l, n ∈ {0, 1,...} and r ∈ Z, the normalized cyclotomic ψ-coefficient n: = p− r l,pa ⌊ n−p a−1 −lp a p a−1 (p−1) k≡r (mod p a) (−1) k ( n) ( k−r p k
A UNIFIED THEORY OF ZERO-SUM PROBLEMS, SUBSET SUMS AND COVERS OF Z
, 2004
"... Zero-sum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdős and investigated by many researchers. In an earlier ..."
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Cited by 3 (3 self)
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Zero-sum problems on abelian groups, subset sums in a field and covers of the integers by residue classes, are three different active topics initiated by P. Erdős and investigated by many researchers. In an earlier
BINOMIAL COEFFICIENTS AND THE RING OF p-ADIC INTEGERS
"... Abstract. Let k> 1 be an integer and let p be a prime. We show that if pa � k < 2pa or k = paq + 1 (with q < p/2) for some a = 1, 2, 3,..., then the set { ( n) : n = 0, 1, 2,...} is dense in the ring Zp of p-adic integers, k i.e., it contains a complete system of residues modulo any power ..."
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Cited by 3 (1 self)
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Abstract. Let k> 1 be an integer and let p be a prime. We show that if pa � k < 2pa or k = paq + 1 (with q < p/2) for some a = 1, 2, 3,..., then the set { ( n) : n = 0, 1, 2,...} is dense in the ring Zp of p-adic integers, k i.e., it contains a complete system of residues modulo any power of p. 1.
