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The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields

Takashi Fukuda, Hisao Taya (1995)

Acta Arithmetica

1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension k ∞ of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of k ∞ / k . We know by the Ferrero-Washington...

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

Consider a recurrence sequence ( x k ) k ∈ ℤ of integers satisfying x k + n = a n - 1 x k + n - 1 + . . . + a ₁ x k + 1 + a ₀ x k , where a ₀ , a ₁ , . . . , a n - 1 ∈ ℤ are fixed and a₀ ∈ -1,1. Assume that x k > 0 for all sufficiently large k. If there exists k₀∈ ℤ such that x k ₀ < 0 then for each negative integer -D there exist infinitely many rational primes q such that q | x k for some k ∈ ℕ and (-D/q) = -1.

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