Unpublished notes

  1. Hilbert's inequality (pdf)
  2. The Byrnes-Gauger theorem (pdf) (replace line -1 on page 2 by correction)
  3. An inequality for the determinant of a positive definite Hermitian matrix (pdf)
  4. Short solutions of AX=B using a LLL-based Hermite normal form algorithm (pdf)
  5. Generalised eigenspaces and the block upper triangular form (pdf)
  6. Short multipliers for the extended gcd problem (pdf )
  7. On the LCM of n positive integers, John Campbell (written up by K.R. Matthews) (pdf)
  8. Some continued fraction identities associated with √D (pdf)
  9. Primitive Pythagorean triples and the negative Pell equation (pdf)
  10. On the continued fraction expansion of √22n+1 (pdf)
  11. A unimodular matrix and Pell's equation (pdf)
  12. Reduced quadratic irrationals and Pell's equation (pdf)
  13. Solving Ax2 - By2 = N in integers, where A > 0, B > 0, N ≠ 0 and D = AB is not a perfect square and gcd(A,B) = gcd(A,N) = 1 (pdf)
  14. A nearest integer formula for [x/m], x real, m a non-zero integer (pdf)
  15. On the definition of nearest integer reduced quadratic surd (with John Robertson).
  16. Converting the regular continued fraction of √D to the nearest integer continued fraction
  17. A class of generalized 3x+1 mappings of Benoit Cloitre (pdf)
  18. An extension of the Hermite-Serret algorithm (pdf) (updated 1st October 2012)
  19. Solving the generalised Pell equation ax2-by2=±1
  20. Pseudocode for finding the shortest multipliers for the extended gcd problem (18th August 2011)
  21. Some remarks about the solutions of x2-Dy2=±N
  22. A note on the Markoff numbers conjecture (last modified 5th November 2013)
  23. On a transformation of Lagrange for solving ax2 + bxy + cy2 = n, where b2–4ac < 0
  24. An equivalent form of the Dujella unicity conjecture
  25. Geometric nature of an equivalence class for the diophantine equation x2 – Dy2 = N, D > 0 and non square, N > 0
  26. Geometric nature of an equivalence class for the diophantine equation x2 – Dy2 = N, D > 0 and non square, N < 0
  27. Intermediate convergents
  28. On Markov's doubly-infinite sequence of forms
  29. On the continued fractions of conjugate quadratic irrationalities
  30. Solving the Diophantine equation ax2+bxy+cy2+dx+ey+f=0, (the hyperbola case was corrected 5th January 2020)
  31. Hardy and Wright Theorem 172 when PS-QR=±4 instead of ±1 (13th August 2016)
  32. On the convergents of semi-regular continued fractions (revised 25th November 2010)
  33. Converting the regular continued fraction of √D to the nearest integer continued fraction
  34. Period  two NSCF and NICF expansions of √D (23rd April 2009).

Last modified 6th December 2020