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18
COMBINATORIAL CONGRUENCES Modulo Prime Powers
, 2007
"... Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f(x) ∈ Z[x]. We establish the congruence deg f p ∑ k≡r (mod p α) ..."
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Cited by 25 (15 self)
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Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f(x) ∈ Z[x]. We establish the congruence deg f p ∑ k≡r (mod p α)
A numbertheoretic approach to homotopy exponents of SU(n)
, 2005
"... We use methods of combinatorial number theory to prove that, for each n ≥ 2 and any prime p, some homotopy group πi(SU(n)) contains an element of order p n−1+ordp(⌊n/p⌋!) , where ordp(m) denotes the largest integer α such that p α  m. ..."
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Cited by 21 (18 self)
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We use methods of combinatorial number theory to prove that, for each n ≥ 2 and any prime p, some homotopy group πi(SU(n)) contains an element of order p n−1+ordp(⌊n/p⌋!) , where ordp(m) denotes the largest integer α such that p α  m.
Combinatorial congruences and ψoperators
 Finite Fields Appl
"... The ψoperator for (ϕ,Γ)modules plays an important role in the study of Iwasawa theory via Fontaine’s big rings. In this note, we prove several sharp estimates for the ψoperator in the cyclotomic case. These estimates immediately imply a number of sharp padic combinatorial congruences, one of whi ..."
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Cited by 7 (3 self)
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The ψoperator for (ϕ,Γ)modules plays an important role in the study of Iwasawa theory via Fontaine’s big rings. In this note, we prove several sharp estimates for the ψoperator in the cyclotomic case. These estimates immediately imply a number of sharp padic combinatorial congruences, one of which extends the classical congruences of Fleck (1913) and Weisman (1977). 1 Combinatorial Congruences Let p be a prime, n ∈ Z>0. Throughout this paper, let [x] denote the integer part of x if x ≥ 0 and [x] = 0 if x < 0. In the author’s course lectures [4] on Fontaine’s theory and padic Lfunctions given at UC Irvine (spring 2005) and at the Morningside Center of Mathematics (summer 2005), the following two congruences were discovered. Theorem 1.1. For integers r ∈ Z, j ≥ 0, we have k≡r(modp) (−1) n−k � n k � � k−r
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
LUCASTYPE CONGRUENCES FOR CYCLOTOMIC ψCOEFFICIENTS
 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 ψcoefficient 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 ψcoefficient 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 GENERALIZATION OF THE GAUSSIAN FORMULA AND A QANALOG OF FLECK’S CONGRUENCE
"... ar ..."
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Fleck’s congruence, associated magic squares and a zeta identity, Funct
 Approx. Comment Math
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(−1) k
, 2007
"... Abstract. Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f(x) ∈ Z[x]. We establish the congruence deg f p � k≡r (mod pα � n ..."
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Abstract. Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f(x) ∈ Z[x]. We establish the congruence deg f p � k≡r (mod pα � n
Combinatorial Congruences and A~Operators
, 2005
"... Abstract The A~operator for (', \Gamma)modules plays an important role in thestudy of Iwasawa theory via Fontaine's big rings. In this note, we prove several sharp estimates for the A~operator in the cyclotomic case.These estimates immediately imply a number of sharp padic combinatori ..."
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Abstract The A~operator for (', \Gamma)modules plays an important role in thestudy of Iwasawa theory via Fontaine's big rings. In this note, we prove several sharp estimates for the A~operator in the cyclotomic case.These estimates immediately imply a number of sharp padic combinatorial congruences, one of which extends the classical congruences of
n k
, 2008
"... The ψoperator for (ϕ, Γ)modules plays an important role in the study of Iwasawa theory via Fontaine’s big rings. In this note, we prove several sharp estimates for the ψoperator in the cyclotomic case. These estimates immediately imply a number of sharp padic combinatorial congruences, one of wh ..."
Abstract
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The ψoperator for (ϕ, Γ)modules plays an important role in the study of Iwasawa theory via Fontaine’s big rings. In this note, we prove several sharp estimates for the ψoperator in the cyclotomic case. These estimates immediately imply a number of sharp padic combinatorial congruences, one of which extends the classical congruences of Fleck (1913) and Weisman (1977). 1 Combinatorial Congruences Let p be a prime, n ∈ Z>0. Throughout this paper, let [x] denote the integer part of x if x ≥ 0 and [x] = 0 if x < 0. In the author’s course lectures [4] on Fontaine’s theory and padic Lfunctions given at UC Irvine (spring 2005) and at the Morningside Center of Mathematics (summer 2005), the following two congruences were discovered. Theorem 1.1. For integers r ∈ Z, j ≥ 0, we have k≡r(modp) (−1) n−k