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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 number-theoretic 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 ψ-operators
- Finite Fields Appl
"... The ψ-operator 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 ψ-operator in the cyclotomic case. These estimates immediately imply a number of sharp p-adic combinatorial congruences, one of whi ..."
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Cited by 7 (3 self)
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The ψ-operator 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 ψ-operator in the cyclotomic case. These estimates immediately imply a number of sharp p-adic 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 p-adic L-functions 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
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 GENERALIZATION OF THE GAUSSIAN FORMULA AND A Q-ANALOG 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 p-adic combi-natori ..."
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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 p-adic combi-natorial congruences, one of which extends the classical congruences of
n k
, 2008
"... The ψ-operator 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 ψ-operator in the cyclotomic case. These estimates immediately imply a number of sharp p-adic combinatorial congruences, one of wh ..."
Abstract
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The ψ-operator 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 ψ-operator in the cyclotomic case. These estimates immediately imply a number of sharp p-adic 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 p-adic L-functions 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
