Publications

  Note that some of the file versions here may differ from the final version.  Look for corrections at the bottom of the page for the ones that are in colour.    

1. A note of Dujella's Conjecture, with M. Le, Glasnik Matematicki, 58, no. 1 (2023), 59-65, Dujella 

2. Infinitely many sign changes of the Liouville function on  quadratics, Proc. Amer. Math. Soc. 150 (2022) 3799-3809 Liouville  

3. The Markoff-Fibonacci Numbers, The Fibonacci Quarterly, 58, no. 5 (2021), p. 222-228  MF.pdf    

4. Markov equation with Pell components, with Bir Kafle and Alain Togbé, The Fibonacci Quarterly, 58, no. 3 (August  2021), p. 226-230 KST2   

5. A complete classification of well-rounded real quadratic ideal lattices, Journal of number theory, 207, February 2020, 349-355 lattices.pdf 

6. New upper bounds for Ramanujan primes, with Pablo Arés, Glasnik Matematicki, 53, no. 1 (2018), 1-7 ramaprimes3.pdf

7. Markov equation with Fibonacci components, with Florian Luca, The Fibonacci Quarterly, 56, no. 2 (May 2018), p. 126  MF1.pdf 

8. Extending theorems of Serret and Pavone, with Keith Matthews and John Robertson, Journal Of Integer Sequences, 20, Article 17.10.5 (2017)  KJA.pdf

9. An improved upper bound for Ramanujan primes, Integers, 15, no. A52 (2015), ramaprimes2.pdf

10. On the fundamental solutions of binary quadratic form equations, with J. Robertson and K. Matthews, Acta Arithmetica, 169 (2015), 291-299  bqf.pdf

11. Sign changes of the Liouville function on some irreducible quadratic polynomials, Journal of Combinatorics and Number Theory, 7.1 (2015)  Liouville.pdf 

12. D(-1)-quadruples and products of two primes, Glasnik Matematicki, 50, no. 2 (2015) 2primes.pdf

13. An upper bound for Ramanujan primes, Integers, 14, no. A19 (2014), ramaprimes.pdf

14. On the prime divisors of elements of a D(-1) quadruple, Glasnik Matematicki, 49, no. 2 (2014), quadruples.pdf

15. ¿Debemos intentar resolver la conjetura de Markoff?, La Gaceta de la RSME, Vol. 16 (2013), Núm. 2, Págs. 313–330, Markoff.pdf (Translation into English: Should we try to solve the Markoff conjecture? English.pdf )

16. Class number one criteria for real quadratic fields with discriminant k^2 p^2+-4p, with R. A. Mollin, Journal of Combinatorics and Number theory, 4.1, (2012), 65-79, onekp.pdf

17. Residuacity and genus theory of forms, with R. A. Mollin, Journal of Number theory, 132, no. 1, (2012), 103-116,resgenus.pdf

18. An improvement of the Minkowski bound for quadratic orders using the Markoff theorem, Journal of Number theory,131, no. 8, 1420-1428 (2011),   minkmar

19. Euler Rabinowitsch polynomials and class number problems revisited, with R. A. Mollin, Funct. Approx. Comment.Math, 45, no. 2 (2011), 271-288.

20. Pell equations: non-principal Lagrange criteria and central norms, with R.A. Mollin, Canadian Mathematical bulletin, 55 (2012), 774-782, centralnorms.pdf

21. Central norms: Applications to Pell's equation, with R.A. Mollin, Far East journal of Mathematical Sciences, 38, no. 2, (2010),  225--252, pellappl.pdf

22. A note on residuacity and criteria for prime representation, with R.A. Mollin, JP journal of Algebra, Number Theory and Applications, 16 , no. 2, (2010), 153--159, rescrit.pdf

23. A note on the negative Pell equation, with R. A. Mollin, International journal of Algebra, 4, no. 19 (2010), 919-922.noteonpell.pdf

24. Markoff numbers and ambiguous classes, Journal de Théorie des Nombres de Bordeaux,  21, no. 3 (2009), 755-768.  markamb.pdf

25. A note on the Markoff conjecture, Biblioteca de la Revista Matemática Iberoamericana, Proceedings of the  Segundas Jornadas de Teoría de Números (Madrid, 2007), pp. 253-260

26.  Generalized Lebesgue-Ramanujan-Nagell Equations, with N. Saradha,  Proceedings of the International conference on Diophantine Equations, TIFR, Mumbai, India, (2008) pp. 207-223.   conf.pdf

27.  Solutions of some generalized Ramanujan-Nagell equations via binary quadratic forms, with N. Saradha,  Publ. Math.  Debrecen,  71, no. 3-4 (2007), pp. 349-374. deb.pdf 

28. Solutions of some generalized Ramanujan-Nagell equations, with N. Saradha,  Indag. Math. (N.S.), 17 (1) (2006), 103-114.  indag.pdf

29. Prime producing quadratic polynomials and class number one or two, The Ramanujan Journal, 10, No. 1 (2005), pp 5-22.  

30. Prime producing polynomials: Proof of  a conjecture by  Mollin and Williams,  Acta Arithmetica,  89, no 1 (1999) pp 1-7

31.  Computations of  class numbers of real quadratic fields, ccc.pdf,  67, no 223 (1998) pp 1285-1308.

Corrections

• An improvement of the Minkowski bound for quadratic orders using the Markoff theorem, Journal of Number theory,131, no. 8, 1420-1428 (2011). (see link above.)

1. In lemma 3.3 in statement 1, the first triple given is not necessarily ordered. Given (c_1, c_2, c) is an ordered Markoff triple, the correct statement 1 here would be:

2. Either (3c_1c_2-c, c_1, c_2)  or (c_1, 3c_1c_2-c, c_2) is ordered. The triples (c_1, c, 3cc_1-c_2) and (c_2, c, 3cc_2-c_1) are ordered.

3. As a result in the proof of Lemma 3.5, at the very end we would have to consider also the triple (c_2, 3cc_2-c_1, c) if this is ordered. 

• The Markoff-Fibonacci Numbers, The Fibonacci Quarterly, 58, no. 5 (December 2020), p. 222 

In section 2.2 the statement on Hurwitz's result  states that he showed that xx_3 . . . x_n ≤ n. This is a mis-quote. He proves that this is true for a fundamental solution.  In the case of the Markoff equation this would mean that x_3=1 and n=3 and hence x≤3.