On the Chinese Remainder Theorem and Its Applications

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On the Chinese Remainder Theorem and Its
Applications
Tapia, Carmelo Tello, David Seniors. University
of Illinois at Chicago
Faculty Mentors Kutzko, Phil and Madison,
Eugene Professors at the University of
Iowa Summer Research Opportunity Program
2000 July 28, 2000. Iowa City, Iowa
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Introduction In a book similar to that of the
Arithmetic in Nine Sections, (1257 AD) written
by the Chinese mathematician, Sun-tzï, we
encounter the first Chinese problem in
indeterminate analysis. The problem says There
are things of an unknown number which when
divided by 3 leaves 2, by 5 leave 3, and by 7
leave 2. What is the (smallest) number? This
problem is considered to be the beginnings of the
famous Chinese Remainder Theorem of Elementary
Number Theory. In our process of extending the
Chinese Remainder Theorem to polynomials, we
found that in the particular case when the
divisors are different prime polynomials of
degree 1, the algorithm for finding the desired
polynomial is the LaGrange Interpolation Formula
found in Numerical Analysis.
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The Chinese Remainder Theorem. Let m1, m2, , mr
denote r positive integers that are relatively
prime in pairs, and let a1, a2, , ar denote any
r integers. Then the congruences x? a1 (mod
m1), x? a2 (mod m2), , x? ar (mod mr) have
common solutions. If x0 is one such solution,
then an integer x satisfies the congruences x? a1
(mod m1), x? a2 (mod m2), , x? ar (mod mr) if
and only if x is of the form x x0km for some
integer k. Here m m1m2mr. Furthermore, the
solution x is unique modulo m m1m2mr.
Example Given x?2 (mod 3), x?3 (mod 5) and
x?2 (mod 7). Find such a x? By the theorem of
Linear Congruences, we know that there exists
integers y1, y2, and y3 such 35y1?1 (mod 3),
21y2?1 (mod 5), and 15y3?1 (mod 7). The
Euclidean algorithm gives us that y1-1, y21 and
y31 respectively. Also, we have that m
(3)(5)(7)105. Then, x(35)(-1)(2)21(1)(3)15(1)
(2)23. Thus, an Algorithm for finding x0 is
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Chinese Remainder Theorem for Polynomials. Let
m1(x), m2(x), , mr(x) denote r prime polynomials
of degree p?1 that are relatively prime in pairs,
and let b1(x), b2(x), , br(x) denote any r prime
polynomials of degree at most p-1. Then the
system of congruences P(x)? bi(x) mod mi(x),
i1, 2, , r has a unique solution modulo g(x),
where g(x) m1m2mr. Special case of CRT for
Polynomials (Lagrange Interpolation Formula).
Let m1(x), m2(x), , mr(x) denote r prime monic
polynomials of degree 1 that are relatively prime
in pairs, and let b1, b2, , br denote any r
prime polynomials of degree 0. Then the system
of congruences P(x)? bi mod mi(x), i1, 2, , r
has a unique solution modulo g(x), where g(x)
m1m2mr.
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Example Assume you have a polynomial that when
it is divided by (x-1) you get remainder 3, when
it is divided by (x-2) you get remainder 2 and
when it is divided (x-3), you get remainder 1.
Find such a polynomial? Using the CRT for
polynomials we get that g(x)(x-1)(x-2)(x-3). Let
m1(x)x-1, m2(x)x-2, and m3(x)x-3. Now let
b13, b22, and b3-1. Using the algorithm from
the table, we find that ½, -1, and ½ are the
inverses of g(x)/(x-ai) for i1, 2 and 3,
respectively. Thus P(x)(x-2)(x-3)(1/2)(3)(x-1)(
x-3)(-1)(2)(x-1)(x-2)(1/2)(-1)-x22x2
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A Historical Remark The Chinese Remainder Theorem
was first presented as problem 26 of the last
volume of Master Suns Mathematical Manual, which
divides into three volumes, sometime before 1213
(?). Joseph Lagrange presented his interpolation
formula, which is described by him as a short
version of Isaac Newtons (1642-1727)
interpolation formula in his Lectures at the
Ecole Normale in 1795. According to 2, many of
the Chinese findings n mathematics ultimately
made their way to Europe via India and Arabia.
The Chinese Remainder Theorem became known in
Europe through article, Jottings on the science
of Chinese arithmetic, by Alexander Wylie in
1853 1. Furthermore, 1 says that J.L.
Lagrange worked on problems on Indeterminate
Analysis around 1767-68. Whether there was any
direct transmission of mathematical knowledge
from China to the West remains a matter of
conjecture. However, the possibility should not
be dismissed out of hand, as many historians of
mathematics are inclined to do either because
they find the idea unpalatable or because there
is insufficient documentary evidence. The fact
remains that, as early as the third century B.C.
Chinese silk and fine ironware were to be found
in the markets of Imperial Rome. And a few
centuries later a whole range of technological
innovations found their way slowly to Europe. It
is not unreasonable to argue that some of Chinas
intellectual products, including mathematical
knowledge, were also carried westwards to Europe,
there perhaps to remain dormant during Europes
intellectual Dark Ages, but coming to life once
more with the cultural awakening of the
Renaissance 3.
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References 1. Dickson, Leonard. History of the
Theory of Numbers, Volume II. Chelsea Publishing
Company, 1992. pg. 57-64. 2. Eves, Howard. An
Introduction to the History of Mathematics.
Saunders College Publishing, 1992. pg. 212-9. 3.
Gheverghese, George. The Crest of the Peacock,
Non-European Roots of Mathematics. Penguin
Books, 1992. pg. 204-214. 4. Goldstinen, Herman
H. A History of Numerical Analysis From the 16 th
through the 19 th Century. Springer-Verlag,
1977. pg. 68-71, 171. 5. Hungerford, Thomas.W.
Algebra. Springer-Verlag, 1974. pg. 131-2. 6.
Lagrange, Joseph Louis. From the French by Thomas
J. McCormack. Elementary Mathematics. Chicago,
1901. pg. 146-9. 7. Lang, Serge. Algebra.
Addison-Wesley, 1967. pg 63-4. 8. Niven, Ivan
Zuckerman, H.S. An Introduction to the Theory of
Numbers. John Wiley Sons, 1980. pg. 44-5. 9.
Silverman, Joseph H. A Friendly Introduction to
Number Theory. Prentice Hall, 1997. pg.
67-8. 10. Yan, Li, Shiran Du. Translated by John
N. Crossley and Anthony W. C. Lun. Chinese
Mathematics, A Concise History. Clarendon Press,
1987. pg. 93, 161-6.
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Acknowledgements

• Special thanks to The University of Iowa
• Graduate College/Summer Research Opportunity
  Program Staff.
• Graduate Students at the Math Department.
• Mathematics Department in particular Professors
  Daniel Anderson, Phil Kutzko, and Eugene Madison.
• For hosting us during the progress of the paper.
• Special thanks to The University of Illinois at
  Chicago
• Illinois Louis Stokes Alliance for Minority
  Participation
• Mathematics, Statistics and Computer Sciences
  Department
• Ronald E. McNair Scholar's Post-baccalaureate
  Program
• Rafael Cintrón-Ortiz Latino Cultural Center
• For providing the financial support for assisting
  to the AMS-MAA Conference.