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We record a few observations on number theoretic aspects of Milnor-Witt K-theory, focusing on generalizing classical results on reciprocity laws, Hasse's norm theorem and K_2 of number fields and rings of integers.

We find all solutions to the parametrized family of norm-form equations x^3-(t^3-1)y^3+3(t^3-1)xy+(t^3-1)^2 = \pm 1studied by Amoroso, Masser and Zannier. Our proof relies upon an appeal to lower bounds for linear forms in logarithms and various elementary arguments.

In 1848 Ch.~Hermite asked if there exists some way to write cubic irrationalities periodically. A little later in order to approach the problem C.G.J.~Jacobi and O.~Perron generalized the classical continued fraction algorithm to the three-dimensional case, this algorithm is called now the Jacobi-Perron algorithm. This algorithm is known to provide periodicity only for some cubic irrationalities. In this paper we introduce two new algorithms in the spirit of Jacobi-Perron algorithm: the heuristi...

A positive integer is called an E_jnumber if it is the product of jdistinct primes. We prove that there are infinitely many triples of E_2numbers within a gap size of 32and infinitely many triples of E_3numbers within a gap size of 15 Assuming the Elliot-Halberstam conjecture for primes and E_2numbers, we can improve these gaps to 12and 5 respectively. We can obtain even smaller gaps for almost primes, almost prime powers, or integers having the same exponent pattern i...

In this article, we provide a relation between the \muinvariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic \mathbb{Z}_pextension to a \mathbb{Z}_p^2extension over an imaginary quadratic field. Furthermore we show that the supersingular \mathfrak{M}_H(G)conjecture is equivalent to the fact that the \muinvariants doesn't change as we go up the tower.

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