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### On blocks of arithmetic progressions with equal products

Journal de théorie des nombres de Bordeaux

Let $f\left(X\right)\in ℚ\left[X\right]$ be a monic polynomial which is a power of a polynomial $g\left(X\right)\in ℚ\left[X\right]$ of degree $\mu \ge 2$ and having simple real roots. For given positive integers ${d}_{1},{d}_{2},\ell ,m$ with $\ell <m$ and gcd$\left(\ell ,m\right)=1$ with $\mu \le m+1$ whenever $m<2$, we show that the equation $f\left(x\right)f\left(x+{d}_{1}\right)\cdots f\left(x+\left(\ell k-1\right){d}_{1}\right)=f\left(y\right)f\left(y+{d}_{2}\right)\cdots f\left(y+\left(mk-1\right){d}_{2}\right)$ with $f\left(x+j{d}_{1}\right)\ne 0$ for $0\le j<\ell k$ has only finitely many solutions in integers $x,y$ and $k\ge 1$ except in the case $m=\mu =2,\ell =k={d}_{2}=1,f\left(X\right)=g\left(X\right),x=f\left(y\right)+y.$

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### Number of solutions of cubic Thue inequalities with positive discriminant

Acta Arithmetica

Let F(X,Y) be an irreducible binary cubic form with integer coefficients and positive discriminant D. Let k be a positive integer satisfying $k<\left({\left(3D\right)}^{1/4}\right)/2\pi$. We give improved upper bounds for the number of primitive solutions of the Thue inequality $|F\left(X,Y\right)|\le k$.

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### Some infinite products with interesting continued fraction expansions

Journal de théorie des nombres de Bordeaux

We display several infinite products with interesting continued fraction expansions. Specifically, for various small values of $k$ necessarily excluding $k=3$ since that case cannot occur, we display infinite products in the field of formal power series whose truncations yield their every $k$-th convergent.

Acta Arithmetica

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