Andrej Dujella: Number Theory - Errata et addenda

List of corrections and additional comments to the book Number Theory, Školska knjiga, 2021

2.3 Prime numbers

Page 37, line -8: Instead of "(1601-1665)" it should probably be "(1607-1665)" (see

3.9 Pseudoprimes

Page 76, line -9: Instead of "not divisible by p" it should be "not divisible by pi". (Thanks to Sunben Chiu.)

4.4 Jacobi symbol

Page 98, line 12: In the first product, instead of p it should be P.

4.5 Divisibility of Fibonacci numbers

Page 100, line 18: Instead of "Example 4.12" it should be "Proposition 4.12". (Thanks to Alejandra Alvarado.)

4.6 Exercises

Page 105, line -5: Third part of Exercise 16 should be "c)" instead of "b)". (Thanks to Alejandra Alvarado.)

Page 106, line -11: In Exercise 25.a), instead of Fn+1m it should be Fn+1m-1. (Thanks to Alejandra Alvarado.)

5.3 Sums of four squares

Page 122, line 10: Instead of "modulo p" it should be "modulo l". (Thanks to Clemens Fuchs.)

7.4 Dirichlet characters

Page 182, line -3: Instead of "residue" it should be "nonresidue".

8.8 Simultaneous approximations

Page 234, line 16: Instead of "j = 1...,m" it should be "j = 1,...,m". (Thanks to Tomislav Pejković.)

8.9 LLL algoritam

Page 241, line -6: Instead of "||b1||" it should be "b1". (Thanks to Tomislav Pejković.)

10.3 Pell's equation

Page 287, line 3: Instead of (x1-dy12)n it should be (x12-dy12)n. (Thanks to Alejandra Alvarado.)

Page 292, line 18: Instead of ((x1+y1d)/2)2 it should be (x1+y1d)2/2. (Thanks to Clemens Fuchs.)

10.9 Exercises

Page 329, line -3: In Exercise 15, instead of n ≠ 3 it should be n ≥ 3. (Thanks to Alejandra Alvarado.).

Page 332, line 2: In Exercise 36, instead of 5 · (-1)n it should be 5 · (-1)n+1. (Thanks to Alejandra Alvarado.).

11.4 Polynomial decomposition

Page 355, line -15: Instead of "from the second" it should be "from the third". (Thanks to Mihai Cipu.)

12.1 Quadratic fields

Page 371: Last sentence in Corollary 12.8. should be:
If x1+y1d is the fundamental solution of Pell's equation x2 - dy2 = 1, then x1+y1d = (a + bd)ν, where ν ∈ {1,2}, or x1+y1d = ((a + bd)/2)ν, where ν ∈ {3,6}.
(Thanks to Clemens Fuchs.)

13.2 Roth's theorem

Page 406, line 17: Instead of "|x1" it should be "|x1|". (Thanks to Tomislav Pejković.)

13.4 Approximation by quadratic irrationals

Page 417: It is proved in
J. Schleischitz, Diophantine approximation in prescribed degree, Mosc. Math. J. 18 (2018), 491-516
that in Theorem 13.11, algebraic numbers β can be taken of degree precisely 2.

14.1 Thue equations

Page 434, last line: The solution (-1,1) should be added. (Thanks to Alejandra Alvarado.)

14.3 Linear forms in logarithms

Page 441, lines 15 and 17: Instead of "αn+1" it should be "log αn+1".

14.4 Baker-Davenport reduction

Page 447, line -4: Instead of "a ∈ <0, 1>" it should be "a ∈ (0, 1)".

Page 450, line 5: Instead of "742265900639684191" it should be "742265900639684111". (Thanks to Tomislav Pejković.)

14.6 Diophantine m-tuples

Page 456, line 3: Instead of "polynomial" it should be "polynomials". (Thanks to Matija Kazalicki.)

Page 462: lines 1-5: Let us mention that in the paper
K. Chakraborty, S. Gupta, A. Hoque, On a conjecture of Franusic and Jadrijevic: Counter-examples, Results Math. 78 (2023), Article 18,
it is shown that in certain rings of the form Z[√4k+2] there are elements z which are not difference of two squares but there exist a D(z)-quadruple (explicit examples are given for z = 26 + 6√10 in Z[√10] and z = 18 + 2√58 in Z[√58]).

Page 462:
The current number of references on the web page [126] is 532.

15.1 Introduction to elliptic curves

Page 469: Elementary proofs of the associative law can be found in the papers:
S. Friedl, An elementary proof of the group law for elliptic curves, Groups Complex. Cryptol. 9 (2017), 117-123.
S. Zwegers, On the associativity of the addition on elliptic curves, preprint, 2024.

15.4 Canonical height and Mordell-Weil theorem

Page 503: In Example 15.8, instead of h^(2nP) it should be h(2nP) (twice).

15.5 Rank of elliptic curves

Page 512, line -10: Instead of b'i it should be b˜'i. (Thanks to Tomislav Pejković.)

Page 515, line 19: Instead of "rank 14" it should be "rank ≥ 14".

Page 519:
In the table on rank-record curves, for the torsion group Z/8Z it should be added Voznyy (2021),
for the torsion group Z/9Z it should be added Dujella, Petričević & Rathbun (2022),
and for the torsion group Z/2Z × Z/8Z it should be added AttarBashi, Rathbun & Voznyy (2022), AttarBashi, Fisher, Rathbun & Voznyy (2022), AttarBashi, Fisher & Voznyy (2022), and Rathbun (2003, 2006) should be replaced by Rathbun (2003, 2006, 2013).
In the same table, for torsion groups Z/7Z and Z/2Z × Z/4Z, instead of Klagsbrung it should be Klagsbrun. (Thanks to Maksym Voznyy.)
The details of all record curves can be found on the web page [127]

Page 519: Additional families with rank ≥ 3 and torsion groups Z/8Z and Z/2Z × Z/6Z, and experiments which suggest that there might be infinitely many curves with such torsion groups and rank ≥ 4, can be found in the paper
A. Dujella, M. Kazalicki, J. C. Peral, Elliptic curves with torsion groups Z/8Z and Z/2Z × Z/6Z, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 115 (2021), Article 169, (24pp).

15.7 Elliptic curves over finite fields

Page 533, line -4; page 534, lines 1 and 3: Instead of "E(Fq)" it should be "E(Fq)" (algebraic closure of Fq). (Thanks to Marin Varivoda.)

15.8 Applications of elliptic curves in cryptography

Page 544, line 12: In August 2022, Castryck and Decru proposed an efficient attack to SIDH in the paper
W. Castryck, T. Decru, An efficient key recovery attack on SIDH (preliminary version), Cryptology ePrint Archive, Report 2022/975, 2022.

15.10 Elliptic curve factorization method

Page 552, line 8: In July 2021, Voznyy found an elliptic curve over quadratic field Q(√190) with rank equal to 2 and torsion group Z/15Z. Thus, for each of the 26 groups from (15.43), there is an elliptic curve over a quadratic field with that torsion group and rank ≥ 2. The current rank records for all 26 groups can be found on the web page

16.5 Algorithm for solving Thue equations

Page 569, line 3: Note that since g is irreducible and s ≥ 1, the only roots of unity belonging to K are ± 1.

Page 571, line 4: Instead of β(i1) it should be βi1; instead of μ(i1) it should be μi1; instead of β(ir) it should be βir; instead of μ(ir) it should be μir.

16.7 Diophantine m-tuples and elliptic curves

Page 581, line 14: The parametrization of rational Diophantine triples is published in the paper
M. Kazalicki, B. Naskrecki (with an appendix by L. Lasić), Diophantine triples and K3 surfaces, J. Number Theory 236 (2022), 41-70.


Reference [4] is published as:
N. Adžaga, On the size of Diophantine m-tuples in imaginary quadratic number rings, Bull. Math. Sci. 11(1) (2021) 1950020 (10 pages).

Reference [53] is published as:
N. C. Bonciocat, M. Cipu, M. Mignotte, There is no Diophantine D(-1)-quadruple, J. London Math. Soc. 105 (2022), 63-99.

Reference [67] is publishes as:
Y. Bugeaud, A. Dujella, W. Fang, T. Pejković, B. Salvy, Absolute root separation, Experiment. Math. 31 (2022), 806-813.

In reference [142], the title should be: Rational Diophantine sextuples with square denominators.

In reference [143], instead of "2020" it should be "2021".

Reference [144] is published as:
A. Dujella, M. Kazalicki and V. Petričević, D(n)-quintuples with square elements, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 115 (2021), Article 172, (10pp).

Reference [156] is published as:
A. Dujella, J. C. Peral, Construction of high rank elliptic curves, J. Geom. Anal. 31 (2021), 6698-6724.

In reference [217], instead of "Hidry" it should be "Hindry".

Reference [299] should be:
J.-F. Mestre, Construction d'une courbe elliptique de rang ≥ 12, C. R. Acad. Sci. Paris Ser I Math. 295 (1982), 643-644.

In reference [314], instead of "rank = 20" it should be "rank ≥ 20".

In reference [362], instead of "Schutt" it should be "Schütt".

You may send your comments, remarks and suggestions on the book by e-mail to I will be grateful to anyone who points out inaccuracies or errors in the book.

Web page of the book Number Theory Andrej Dujella - home page