Diophantine m-tuples for linear polynomials

DIOPHANTINE m-TUPLES FOR LINEAR POLYNOMIALS. II. EQUAL DEGREES

In this paper we prove the best possible upper bounds for the number of elements in a set of polynomials with integer coefficients all having the same degree, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. Moreover, we prove that there does not exist a set of more than 12 polynomials with integer coefficients and with the...

Diophantine m-tuples for quadratic polynomials

In this paper, we prove that there does not exist a set with more than 98 nonzero polynomials in Z[X], such that the product of any two of them plus a quadratic polynomial n is a square of a polynomial from Z[X] (we exclude the possibility that all elements of such set are constant multiples of a linear polynomial p ∈ Z[X] such that p2|n). Specially, we prove that if such a set contains only po...

On the number of Diophantine m-tuples

A set of m positive integers is called a Diophantine m-tuple if the product of any two of them is one less than a perfect square. It is known that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. On the other hand, there are infinitely many Diophantine m-tuples for m = 2, 3 and 4. In this paper, we derive asymptotic extimates for the numb...

Symmetry classes of polynomials associated with the dihedral group

‎In this paper‎, ‎we obtain the dimensions of symmetry classes of polynomials associated with‎ ‎the irreducible characters of the dihedral group as a subgroup of‎ ‎the full symmetric group‎. ‎Then we discuss the existence of o-basis‎ ‎of these classes‎.

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm