Displaying similar documents to “Division-ample sets and the Diophantine problem for rings of integers”

Diophantine m -tuples and elliptic curves

Andrej Dujella (2001)

Journal de théorie des nombres de Bordeaux

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A Diophantine m -tuple is a set of m positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form y 2 = ( a x + 1 ) ( b x + 1 ) ( c x + 1 ) , where { a , b , c } , is a Diophantine triple. In particular, we consider the elliptic curve E k defined by the equation y 2 = ( F 2 k x + 1 ) ( F 2 k + 2 x + 1 ) ( F 2 k + 4 x + 1 ) , where k ≥ 2 and F n , denotes the n -th Fibonacci number. We prove that if the rank of E k ( 𝐐 ) is equal to one, or k ≤ 50 , then all integer points on E k are given by ( x , y ) ∈ { ( 0 ± 1 ) , ( 4 F 2 k + 1 F 2 k + 2 F 2 k + 3 ± 2 F 2 k + 1 F 2 k + 2 - 1 × 2 F 2 k + 2 2 + 1 2 F 2 k + 2 F 2 k + 3 + 1 } .

Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Laurent Moret-Bailly, Alexandra Shlapentokh (2009)

Annales de l’institut Fourier

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Let K be a one-variable function field over a field of constants of characteristic 0. Let R be a holomorphy subring of K , not equal to K . We prove the following undecidability results for R : if K is recursive, then Hilbert’s Tenth Problem is undecidable in R . In general, there exist x 1 , ... , x n ∈ R such that there is no algorithm to tell whether a polynomial equation with coefficients in ℚ ( x 1 , ... , x n ) has solutions in R .

On the diophantine equation f(x)f(y) = f(z)²

Maciej Ulas (2007)

Colloquium Mathematicae

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Let f ∈ ℚ [X] and deg f ≤ 3. We prove that if deg f = 2, then the diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in ℚ (t). In the case when deg f = 3 and f(X) = X(X²+aX+b) we show that for all but finitely many a,b ∈ ℤ satisfying ab ≠ 0 and additionally, if p|a, then p²∤b, the equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in rationals.