Displaying similar documents to “Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0”

Method of infinite ascent applied on - ( 2 p · A 6 ) + B 3 = C 2

Susil Kumar Jena (2013)

Communications in Mathematics

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In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for ( A , B , C ) of the Diophantine equation - ( 2 p · A 6 ) + B 3 = C 2 for any positive integral values of p when p ≡ 1 (mod 6) or p ≡ 2 (mod 6). For doing this, we will be using a published result of this author in The Mathematics Student, a periodical of the Indian Mathematical Society.

Division-ample sets and the Diophantine problem for rings of integers

Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)

Journal de Théorie des Nombres de Bordeaux

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We prove that Hilbert’s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called set of integers and of an elliptic curve of rank one over K ). We relate division-ample sets to arithmetic of abelian varieties.

A note on the number of S -Diophantine quadruples

Florian Luca, Volker Ziegler (2014)

Communications in Mathematics

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Let ( a 1 , ⋯ , a m ) be an m -tuple of positive, pairwise distinct integers. If for all 1 ≤ i < j ≤ m the prime divisors of a i a j + 1 come from the same fixed set S , then we call the m -tuple S -Diophantine. In this note we estimate the number of S -Diophantine quadruples in terms of | S | = r .