2.3 Prime numbers
Page 37, line -8: Instead of "(1601-1665)" it should probably be "(1607-1665)" (see https://en.wikipedia.org/wiki/Pierre_de_Fermat).
3.9 Pseudoprimes
Page 76, line -9: Instead of "not divisible by p" it should be "not divisible by p_{i}". (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 F_{n+1}^{m} it should be F_{n+1}^{m-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 "||b_{1}||" it should be "b_{1}". (Thanks to Tomislav Pejković.)
10.3 Pell's equation
Page 287, line 3: Instead of (x_{1}-dy_{1}^{2})^{n} it should be (x_{1}^{2}-dy_{1}^{2})^{n}. (Thanks to Alejandra Alvarado.)
Page 292, line 18: Instead of ((x_{1}+y_{1}√d)/2)^{2} it should be (x_{1}+y_{1}√d)^{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 x_{1}+y_{1}√d
is the fundamental solution of Pell's equation x^{2} - dy^{2} = 1,
then x_{1}+y_{1}√d = (a + b√d)^{ν},
where ν ∈ {1,2},
or x_{1}+y_{1}√d = ((a + b√d)/2)^{ν},
where ν ∈ {3,6}.
(Thanks to Clemens Fuchs.)
13.2 Roth's theorem
Page 406, line 17: Instead of "|x_{1}" it should be "|x_{1}|". (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 "α_{n+1}" it should be "log α_{n+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
Page 462:
The current number of references on the web page [126]
https://web.math.pmf.unizg.hr/~duje/dtuples.html
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^{^}(2^{n}P) it should be h(2^{n}P) (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]
https://web.math.pmf.unizg.hr/~duje/tors/tors.html.
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(F_{q})" it should be "E(F_{q})" (algebraic closure of F_{q}). (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. http://eprint.iacr.org/2022/975.
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 http://web.math.pmf.unizg.hr/~duje/tors/torsquad.html.
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 β_{i1}; instead of μ^{(i1)} it should be μ_{i1}; instead of β^{(ir)} it should be β_{ir}; instead of μ^{(ir)} it should be μ_{ir}.
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.
References
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 duje@math.hr. I will be grateful to anyone who points out inaccuracies or errors in the book.
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