H. W. Lenstra [7] introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in [3] and [6]). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves [3] and C. Hsu [6].2010 Mathematics Subject Classification. 11A05.

For a square-free integer t, Byeon [3] proved the existence of infinitely many pairs of quadratic fields Q(√ D) and Q(√ tD) with D > 0 such that the class numbers of all of them are indivisible by 3. In the same spirit, we prove that for a given integer t ≥ 1 with t ≡ 0 (mod 4), a positive proportion of fundamental discriminants D > 0 exist for which the class numbers of both the real quadratic fields Q(√ D) and Q(√ D + t) are indivisible by 3. This also addresses the complement of a weak form of a conjecture of Iizuka in [8]. As an application of our main result, we obtain that for any integer t ≥ 1 with t ≡ 0 (mod 12), there are infinitely many pairs of real quadratic fields Q(√ D) and Q(√ D + t) such that the Iwasawa λ-invariants associated with the basic Z3-extensions of both Q(√ D) and Q(√ D + t) are 0. For p = 3, this supports Greenberg's conjecture which asserts that λp(K) = 0 for any prime number p and any totally real number field K.

Let K/Q be an algebraic number field of class number one and O K be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in O K under the assumption of the abc conjecture for number fields.

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