We prove that there exist infinite Büchi i sequences in some local rings and local fields, with the exception of the ring $ℤ_p$ of p-adic integers. In $ℤ_p$ there are only finite but arbitrarily long Büchi sequences.
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We consider systems of equations of the form $x_i + x_j = x_k$ and $x_i · x_j = x_k$, which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.
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T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every 𝜀 > 0, the gap between two consecutive numbers A $A^{0.4}$ with two exceptions given in Table 2.
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We consider elliptic curves defined over ℚ. It is known that for a prime p > 3 quadratic twists permute the Kodaira classes, and curves belonging to a given class have the same conductor exponent. It is not the case for p = 2 and 3. We establish a refinement of the Kodaira classification, ensuring that the permutation property is recovered by {refined} classes in the cases p = 2 and 3. We also investigate the nonquadratic twists. In the last part of the paper we discuss the number of isogeny classes of curves for given conductors of some special forms. Representative numerical data are given in the tables.
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We discuss some cancellation algorithms such that the first non-cancelled number is a prime number p or a number of some specific type. We investigate which numbers in the interval (p,2p) are non-cancelled.
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Let F be a Galois extension of a number field k with the Galois group G. The Brauer-Kuroda theorem gives an expression of the Dedekind zeta function of the field F as a product of zeta functions of some of its subfields containing k, provided the group G is not exceptional. In this paper, we investigate the exceptional groups. In particular, we determine all nilpotent exceptional groups, and give a sufficient condition for a group to be exceptional. We give many examples of nonnilpotent solvable and nonsolvable exceptional groups.
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