On the maximal difference between an element and its inverse in residue rings
Authors:
Kevin Ford, Mizan R. Khan, Igor E. Shparlinski and Christian L. Yankov
Journal:
Proc. Amer. Math. Soc. 133 (2005), 34633468
MSC (2000):
Primary 11A07, 11N25
Published electronically:
June 8, 2005
MathSciNet review:
2163580
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We investigate the distribution of where
Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on . We also present some heuristic arguments showing that these lower bounds are probably tight, and thus our technique can be a more appropriate tool to study than a more traditional way using exponential sums.
 1.
József
Beck and Mizan
R. Khan, On the uniform distribution of inverses modulo
𝑛, Period. Math. Hungar. 44 (2002),
no. 2, 147–155. MR 1918681
(2003k:11127), http://dx.doi.org/10.1023/A:1019684111647
 2.
E.
Bombieri, J.
B. Friedlander, and H.
Iwaniec, Primes in arithmetic progressions to
large moduli. III, J. Amer. Math. Soc.
2 (1989), no. 2,
215–224. MR
976723 (89m:11087), http://dx.doi.org/10.1090/S08940347198909767236
 3.
Cristian
Cobeli and Alexandru
Zaharescu, On the distribution of the 𝐹_{𝑝}points
on an affine curve in 𝑟 dimensions, Acta Arith.
99 (2001), no. 4, 321–329. MR 1845688
(2003j:11066), http://dx.doi.org/10.4064/aa9942
 4.
K. Ford, The Distribution of Integers with a Divisor in a Given Interval, Preprint, 2004.
 5.
A. Granville, I. E. Shparlinski and A. Zaharescu, On the Distribution of Rational Functions Along a Curve over and Residue Races, Preprint, 2004.
 6.
H.
Halberstam and H.E.
Richert, Sieve methods, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], LondonNew York, 1974. London
Mathematical Society Monographs, No. 4. MR 0424730
(54 #12689)
 7.
Richard
R. Hall and Gérald
Tenenbaum, Divisors, Cambridge Tracts in Mathematics,
vol. 90, Cambridge University Press, Cambridge, 1988. MR 964687
(90a:11107)
 8.
KarlHeinz
Indlekofer and Nikolai
M. Timofeev, Divisors of shifted primes, Publ. Math. Debrecen
60 (2002), no. 34, 307–345. MR 1898566
(2003b:11076)
 9.
M. R. Khan, Problem 10736: An Optimization with a Modular Constraint, Amer. Math. Monthly 108 (2001), 374375.
 10.
Mizan
R. Khan and Igor
E. Shparlinski, On the maximal difference between an element and
its inverse modulo 𝑛, Period. Math. Hungar.
47 (2003), no. 12, 111–117. MR 2024977
(2004k:11006), http://dx.doi.org/10.1023/B:MAHU.0000010815.14847.96
 11.
Marian
Vajaitu and Alexandru
Zaharescu, Distribution of values of rational maps on the
𝐹_{𝑝}points on an affine curve, Monatsh. Math.
136 (2002), no. 1, 81–86. MR 1908082
(2003f:11089), http://dx.doi.org/10.1007/s006050200035
 12.
Wen
Peng Zhang, On the difference between an integer and its inverse
modulo 𝑛, J. Number Theory 52 (1995),
no. 1, 1–6. MR 1331760
(96f:11123), http://dx.doi.org/10.1006/jnth.1995.1050
 13.
Zhang
Wenpeng, On the distribution of inverses modulo 𝑛, J.
Number Theory 61 (1996), no. 2, 301–310. MR 1423056
(98g:11109), http://dx.doi.org/10.1006/jnth.1996.0151
 14.
Zhiyong
Zheng, The distribution of zeros of an irreducible curve over a
finite field, J. Number Theory 59 (1996), no. 1,
106–118. MR 1399701
(97f:11066), http://dx.doi.org/10.1006/jnth.1996.0090
 1.
 J. Beck and M. R. Khan, On the Uniform Distribution of Inverses Modulo , Periodica Mathematica Hungarica 44 (2002), 147155. MR 1918681 (2003k:11127)
 2.
 E. Bombieri, J. Friedlander and H. Iwaniec, Primes in arithmetic progressions to large moduli. III, J. Amer. Math. Soc. 2, no. 2 (1989), 215224. MR 0976723 (89m:11087)
 3.
 C. Cobeli and A. Zaharescu, On the Distribution of the points on an Affine Curve in Dimensions, Acta Arithmetica 99 (2001), 321329. MR 1845688 (2003j:11066)
 4.
 K. Ford, The Distribution of Integers with a Divisor in a Given Interval, Preprint, 2004.
 5.
 A. Granville, I. E. Shparlinski and A. Zaharescu, On the Distribution of Rational Functions Along a Curve over and Residue Races, Preprint, 2004.
 6.
 H. Halberstam and H.E. Richert, Sieve Methods, Academic Press, 1974. MR 0424730 (54:12689)
 7.
 R. Hall and G. Tenenbaum, Divisors, Cambridge Tracts in Mathematics 90, Cambridge University Press, 1988. MR 0964687 (90a:11107)
 8.
 K.H. Indlekofer and N. M. Timofeev, Divisors of Shifted Primes, Publ. Math. Debrecen 60 (2002), 307345. MR 1898566 (2003b:11076)
 9.
 M. R. Khan, Problem 10736: An Optimization with a Modular Constraint, Amer. Math. Monthly 108 (2001), 374375.
 10.
 M. R. Khan and I. E. Shparlinski, On the Maximal Difference between an Element and its Inverse modulo , Periodica Mathematica Hungarica 47 (2003), 111117. MR 2024977 (2004k:11006)
 11.
 M. Vajaitu and A. Zaharescu, Distribution of Values of Rational Maps on the points on an Affine Curve, Monathsh. Math. 136 (2002), 8186. MR 1908082 (2003f:11089)
 12.
 W. Zhang, On the Difference between an Integer and Its Inverse Modulo , J. Number Theory 52 (1995), 16. MR 1331760 (96f:11123)
 13.
 W. Zhang, On the Distribution of Inverses Modulo , J. Number Theory 61 (1996), 301310. MR 1423056 (98g:11109)
 14.
 Z. Zheng, The Distribution of Zeros of an Irreducible Curve over a Finite Field, J. Number Theory 59 (1996), 106118. MR 1399701 (97f:11066)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
11A07,
11N25
Retrieve articles in all journals
with MSC (2000):
11A07,
11N25
Additional Information
Kevin Ford
Affiliation:
Department of Mathematics, University of Illinois at UrbanaChampaign, 1409 West Green Street, Urbana, Illinois 61801
Email:
ford@math.uiuc.edu
Mizan R. Khan
Affiliation:
Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226
Email:
khanm@easternct.edu
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email:
igor@ics.mq.edu.au
Christian L. Yankov
Affiliation:
Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226
Email:
yankovc@easternct.edu
DOI:
http://dx.doi.org/10.1090/S0002993905079621
PII:
S 00029939(05)079621
Received by editor(s):
July 16, 2004
Published electronically:
June 8, 2005
Communicated by:
David E. Rohrlich
Article copyright:
© Copyright 2005
American Mathematical Society
