Andrej Dujella: Diophantine m-tuples and Elliptic Curves - Errata et addenda

List of corrections and additional comments to the book Diophantine m-tuples and Elliptic Curves, Springer, 2024

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 564.

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 recond 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.

Page 73: In Table 2.4, in rows corresponding to Z/7Z and Z/9Z, it should be added "Youmbai (2024)".

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

Concering 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 B_n = O(log log |n|) and M_n ≤ (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.: Diophantine tuples and multiplicative structure of shifted multiplicative subgroups. 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.