On Simultaneous Palindromes

A palindrome in base $g$ is an integer $N$ that remains the same when its digit expansion in base $g$ is reversed. Let $g$ and $h$ be given distinct integers $>1$. In this paper we discuss how many integers are palindromes in base $g$ and simultaneously palindromes in base $h$

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 $F(x,y)=b$ with unknowns x,y from A and hyper- and superelliptic equations $f(x)=by^m$ 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 $f(x)=by^m$ 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 $f(x)=by^m$ 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 $f(x)=by^m$ 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