S-unit equations and their applications

On the length of arithmetic progressions in linear combinations of S-units

“…For the theory and applications of such and related equations we refer to [3,4,5,6,7,8], and the references therein. Recently, Hajdu [9], and Jarden and Narkiewicz [10], independently, have investigated arithmetic progressions in the linear combinations of elements from such groups Γ.…”

Irreducibility criteria of Schur-type and Pólya-type

“…(ii) Finally, we are left with the case where there is no β s of the form u + α with α satisfying (15). In this case a simple calculation shows that if g s (x) | H 1 (x) − u 0 then there is no linear polynomial dividing H 1 (x) + u 0 .…”

“…In this case a simple calculation shows that if g s (x) | H 1 (x) − u 0 then there is no linear polynomial dividing H 1 (x) + u 0 . Let now g r (x) | H 1 (x) + u 0 for some r = s. We recall the well-known fact (which can also be readily checked) that if 2 has a divisor different from ±1, ±2 in the ring of integers of an imaginary quadratic number field K then we have K = Q(α) with α satisfying (15). Hence a simple calculation yields that now g r (β s ) ∈ {−2, −1, 1, 2} must be valid.…”

“…In [15,[17][18][19][20] some effective and quantitative versions were also established. For example, it was shown in [15] that under the above assumptions concerning f and g there is an effectively computable positive constant c 1 which depends only on the degree, class number and discriminant of K such that if (20) then g( f (x)) is irreducible over Q.…”

Computing all power integral bases in orders of totally real cyclic sextic number fields

“…Equation (14) can always be reduced to a unit equation in two variables over K (see [7]). In our case, since we deal with relative conjugates over M , the factor corresponding to µ cancels, and the units in this unit equation have 4 factors with unknown exponents.…”