Preface
Page v, line 20: The current number of references on the web page [121] https://web.math.pmf.unizg.hr/~duje/dtuples.html is 586.
1.4 Definitions, Main Problems and Conjectures
Page 13: In August 2026, Ana and Matej Jurasić found an infinite family of counterexamples to Conjecture 1.4.5. The smallest counterexample in their family is a=5, b=115, c=19726599787079129775103071759543738099950374011720. Then {a,b,c} is a Diophantine triple and the equation y2=(ax+1)(bx+1)(cx+1), in addition to the expected solutions with x=0, x=d-=8569332541111894343538801554602513003703076192, x=d+=45410624140523615630392927651668130503572757271903488, has an additional solution x=34307130064485443087135776973119544521652824368 (for which none of the factors ax+1, bx+1, and cx+1 is a square, but the product (ax+1)(bx+1)(cx+1) is a square).
1.5 Generalizations of Diophantine m-Tuples
Page 15, line 20: Instead of 8539880 it should be 9539880. (Thanks to Lawrence Somer.)
2.5 Rank of Elliptic Curves
Page 55: The current record for the rank of elliptic curves over Q is the curve with rank ≥ 29 found by N. Elkies and Z. Klagsbrun in August 2024:
y2 + xy = x3 - 27006183241630922218434652145297453784768054621836357954737385x
+ 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497.
Table 2.1 on p.56 and Table 2.2 on p.71 should be updated accordingly.
Page 71: In Table 2.2, in the row corresponding to Z/4Z, it should be added "Li" and in the row corresponding to Z/9Z, it should be added "Voznyy".
Page 73: In Table 2.4, in the rows corresponding to Z/7Z and Z/9Z, it should be added "Youmbai".
3.6.3 Torsion Group Z/2Z × Z/4Z
Page 128, line -6: (157, 127, 43, 12) should be omitted since it produces the same curve as (55, 31, 142, 15). (Thanks to Maksym Voznyy.)
3.9 Elliptic Curves Induced by Diophantine Triples over Quadratic Fields
Page 155, line 5: Instead of 64t8 it should be 64w8. (Thanks to Maksym Voznyy.)
3.11 Exercises
Concerning Exercise 12 on page 166, let us mention that in the recent paper:
A. Dujella, M. Kazalicki, V. Petričević, Rational Diophantine sextuples with strong pair,
Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 119, Article 36 (2025),
it is shown that there exist infinitely many rational Diophantine sextuples
{a,b,c,d,e,f} with the property that
a2+1 and b2+1 are perfect squares.
5.1 Existence of D(n)-quadruples
Page 282, line 2: } is missing after 20. (Thanks to Mihai Cipu.)
5.2 Bounds for the Size of D(n)-tuples
Page 294, Remark 5.2.13: In the recent paper
C. H. Yip, Improved upper bounds on Diophantine tuples with the property D(n),
Bull. Aust. Math. Soc. 111, 428-432 (2025),
it is shown that Bn = O(log log |n|) and
Mn ≤ (2+o(1)) log |n|.
References
In reference [3], instead of "Preprint (2023). arXiv: 2304.01775" it should be "Ramanujan J. 66, Article 11 (2025)".
Reference [237] should be "Kim, S., Yip, C.H., Yoo, S.: Multiplicative structure of shifted multiplicative subgroups and its applications to Diophantine tuples. Canad. J. Math., to appear."
In reference [360], instead of "Preprint (2023). arXiv: 2312.14450" it should be "Acta Arith. 218, 251-271 (2025)".
Reference [363] should be "Zwegers, S.: An elementary approach to the group law on elliptic curves. Arch. Math. (Basel) 123, 477-486 (2024)"
You may send your comments, remarks and suggestions on the book by e-mail to duje@math.hr. I will be grateful to anyone who points out inaccuracies or errors in the book.
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