114 research outputs found
On Simultaneous Palindromes
A palindrome in base is an integer that remains the same when its
digit expansion in base is reversed. Let and be given distinct
integers . In this paper we discuss how many integers are palindromes in
base and simultaneously palindromes in base
Computational experiences on norm form equations with solutions forming arithmetic progressions
In the present paper we solve the equation
NK/Q(x0 + x1α + x2α2 + ... + xn -1αn -1) = 1
in x0, ... , xn -1 ∈ Z, such that x0, ... , xn -1 is an arithmetic progression, where α is a root of the polynomial xn - a, for all integers 2 ≤ a ≤ 100 and n ≥ 3
Effective results for Diophantine equations over finitely generated domains
Let A be an arbitrary integral domain of characteristic 0 which is finitely
generated over Z. We consider Thue equations with unknowns x,y from
A and hyper- and superelliptic equations with unknowns from A,
where the binary form F and the polynomial f have their coefficients in A,
where b is a non-zero element from A, and where m is an integer at least 2.
Under the necessary finiteness conditions imposed on F,f,m, we give explicit
upper bounds for the sizes of x,y in terms of suitable representations for
A,F,f,b Our results imply that the solutions of Thue equations and hyper- and
superelliptic equations over arbitrary finitely generated domains can be
determined effectively in principle. Further, we generalize a theorem of
Schinzel and Tijdeman to the effect, that there is an effectively computable
constant C such that has no solutions in x,y from A with y not 0 or
a root of unity if m>C. In our proofs, we use effective results for Thue
equations and hyper- and superelliptic equations over number fields and
function fields, some effective commutative algebra, and a specialization
argument.Comment: 37 page
Effective results for hyper- and superelliptic equations over number fields
We consider hyper- and superelliptic equations with unknowns x,y
from the ring of S-integers of a given number field K. Here, f is a polynomial
with S-integral coefficients of degree n with non-zero discriminant and b is a
non-zero S-integer. Assuming that n>2 if m=2 or n>1 if m>2, we give completely
explicit upper bounds for the heights of the solutions x,y in terms of the
heights of f and b, the discriminant of K, and the norms of the prime ideals in
S. Further, we give a completely explicit bound C such that has no
solutions in S-integers x,y if m>C, except if y is 0 or a root of unity. We
will apply these results in another paper where we consider hyper- and
superelliptic equations with unknowns taken from an arbitrary finitely
generated integral domain of characteristic 0.Comment: 31 page
On geometric progressions on Pell equations and Lucas sequence
We consider geometric progressions on the solution set of Pell equations and give upper bounds for such geometric progressions. Moreover, we show how to find for a given four term geometric progression a Pell equation such that this geometric progression is contained in the solution set. In the case of a given five term geometric progression we show that at most finitely many essentially distinct Pell equations exist, that admit the given five term geometric progression. In the last part of the paper we also establish similar results for Lucas sequences
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