Lucas - Type Congruences for Cyclotomic Ψ - Coefficients 3
نویسندگان
چکیده
Let p be any prime and a be a positive integer.
منابع مشابه
Lucas-type Congruences for Cyclotomic Ψ-coefficients
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 r } l,pa := p − ⌊ n−pa−1−lpa pa−1(p−1) ⌋ ∑ k≡r (mod pa) (−1) (n k )( k−r pa l ) is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congruences. This implies that a congruence of Wan i...
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Let p be any prime and a be a positive integer.
متن کاملSe p 20 06 Preprint , arXiv : math . NT / 0512012 . LUCAS - TYPE CONGRUENCES FOR CYCLOTOMIC ψ - COEFFICIENTS
Let p be any prime and a be a positive integer. := p − n−p a−1 −lp a p a−1 (p−1) k≡r (mod p a) (−1) k n k k−r p a l is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congru-ences. This implies that a congruence of Wan is often optimal, and two conjectures of Sun and Davis are true.
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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 Congruen...
متن کاملCongruences for q-Lucas Numbers
For α, β, γ, δ ∈ Z and ν = (α, β, γ, δ), the q-Fibonacci numbers are given by F ν 0 (q) = 0, F ν 1 (q) = 1 and F ν n+1(q) = q αn−βF ν n (q) + q γn−δF ν n−1(q) for n > 1. And define the q-Lucas number Ln(q) = F ν n+1(q)+ q γ−δF ν∗ n−1(q), where ν∗ = (α, β− α, γ, δ − γ). Suppose that α = 0 and γ is prime to n, or α = γ is prime to n. We prove that Ln(q) ≡ (−1) (mod Φn(q)) for n > 3, where Φn(q) i...
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