Bounds for frequencies of residues of second-order recurrences modulo p r

Walter Carlip; Lawrence Somer

Mathematica Bohemica (2007)

  • Volume: 132, Issue: 2, page 137-175
  • ISSN: 0862-7959

Abstract

top
The authors examine the frequency distribution of second-order recurrence sequences that are not p -regular, for an odd prime p , and apply their results to compute bounds for the frequencies of p -singular elements of p -regular second-order recurrences modulo powers of the prime p . The authors’ results have application to the p -stability of second-order recurrence sequences.

How to cite

top

Carlip, Walter, and Somer, Lawrence. "Bounds for frequencies of residues of second-order recurrences modulo $p^r$." Mathematica Bohemica 132.2 (2007): 137-175. <http://eudml.org/doc/250255>.

@article{Carlip2007,
abstract = {The authors examine the frequency distribution of second-order recurrence sequences that are not $p$-regular, for an odd prime $p$, and apply their results to compute bounds for the frequencies of $p$-singular elements of $p$-regular second-order recurrences modulo powers of the prime $p$. The authors’ results have application to the $p$-stability of second-order recurrence sequences.},
author = {Carlip, Walter, Somer, Lawrence},
journal = {Mathematica Bohemica},
keywords = {Lucas; Fibonacci; stability; uniform distribution; recurrence; uniform distribution; recurrence},
language = {eng},
number = {2},
pages = {137-175},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bounds for frequencies of residues of second-order recurrences modulo $p^r$},
url = {http://eudml.org/doc/250255},
volume = {132},
year = {2007},
}

TY - JOUR
AU - Carlip, Walter
AU - Somer, Lawrence
TI - Bounds for frequencies of residues of second-order recurrences modulo $p^r$
JO - Mathematica Bohemica
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 132
IS - 2
SP - 137
EP - 175
AB - The authors examine the frequency distribution of second-order recurrence sequences that are not $p$-regular, for an odd prime $p$, and apply their results to compute bounds for the frequencies of $p$-singular elements of $p$-regular second-order recurrences modulo powers of the prime $p$. The authors’ results have application to the $p$-stability of second-order recurrence sequences.
LA - eng
KW - Lucas; Fibonacci; stability; uniform distribution; recurrence; uniform distribution; recurrence
UR - http://eudml.org/doc/250255
ER -

References

top
  1. A distribution property for linear recurrence of the second order, Proc. Amer. Math. Soc. 50 (1975), 101–106. (1975) Zbl0318.10006MR0369240
  2. Bounds for frequencies of residues of regular second-order recurrences modulo p r , Number Theory in Progress, Vol. 2 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin, 1999, pp. 691–719. (1999) MR1689539
  3. On the numerical factors of the arithmetic forms α n ± β n , Ann. Math. 15 (1913/14), 30–70. (1913/14) MR1502458
  4. An extended theory of Lucas’ functions, Ann. of Math. 31 (1930), 419–448. (1930) MR1502953
  5. Fundamentals of Number Theory, Dover Publications Inc., Mineola, NY, 1996, Reprint of the 1977 original. MR1382656
  6. Théorie des fonctions numériques simplement périodiques, Amer. J. Math. 1 (1878), 184–240, 289–321. (1878) MR1505176
  7. The Calculus of Finite Differences, Macmillan, London, 1951. (1951) MR0043339
  8. A simple and general approach to the decimation of feedback shift-register sequences, Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform. 17 (1988), 327–331. (1988) Zbl0678.10011MR0967952
  9. Maximal frequencies of elements in second-order linear recurring sequences over a finite field, Elem. Math. 46 (1991), 139–143. (1991) MR1119645
  10. An Introduction to the Theory of Numbers, fifth ed, John Wiley & Sons Inc., New York, 1991. (1991) MR1083765
  11. Solution to problem H-377, Fibonacci Quart. 24 (1986), 284–285. (1986) 
  12. Upper Bounds for Frequencies of Elements in Second-Order Recurrences Over a Finite Field, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 527–546. (1993) MR1271393
  13. Stability of second-order recurrences modulo p r , Int. J. Math. Math. Sci. 23 (2000), 225–241. (2000) MR1757803
  14. The arithmetical theory of linear recurring series, Trans. Amer. Math. Soc. 35 (1933), 600–628. (1933) MR1501705
  15. Distribution modulo p h of the general linear second order recurrence, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 92–100. (1975) MR0419375

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.