On the size of Diophantine m-tuples in imaginary quadratic number rings

Abstract

A Diophantine m-tuple is a set of m distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that m 42. Our proof is relatively simple compared to the proofs of the similar results in positive integers.

On the extensions of the Diophantine triples in Gaussian integers

“…Then there are r, s and t in Z[i] such that ab + 1 = r 2 , ac + 1 = s 2 , bc + 1 = t 2 . In [1], the following lemma was proven (for general imaginary quadratic number rings). If there is d ∈ Z[i] such that {a, b, c, d} is a Diophantine quadruple, then there are x, y, z ∈ Z[i] such that ad + 1 = x 2 , bd + 1 = y 2 , cd + 1 = z 2 .…”

“…All the claims are proven inductively. We first prove that the sequence (|W (1) m |) m is increasing. For m = 1, the inequality…”

“…For the sequence (W (1) m ) m defined in Lemma 4.3, its elements W (1) 2m ≡ −2mk + 2m + 1 (mod 4k(k − 1)) and W (1) 2m+1 ≡ (2m + 3)k − 2m − 2 (mod 4k(k − 1)) for all m ∈ N 0 . This sequence (W (1) m ) m is increasing in absolute value, and the inequality |W (1) m | (8|k| 2 − 4|k| − 3) m−1 holds for all m 0. Similarly, the other sequences (W ( j) m ) m , for j = 2, .…”

“…A solution of this system gives us two simultaneous approximations of square roots close to 1. One can use Diophantine approximations in the general case (by assuming that |c| is much bigger than |b|, say |c| > |b| 15 ), which was done in [1]. However, here we show that this is not useful for the triple of the form {k − 1, k + 1, 16k 3 − 4k}.…”

“…we find only about 10 papers solving similar problems in the ring of Gaussian integers. We can highlight [5] and [9], which deal with the extension of Diophantine triples from oneparameter families, and [1], which shows that there is no Diophantine m-tuple in imaginary quadratic number ring with m 43.…”