Introduction to Maple (Math 502)
Assignments
Spring Semester, 2007
End of Semester Schedule
 Wed., May. 16:
 Final Examination: 4:00  6:00 p.m.
 Tue., May. 15:
 Office Hours: 5:30  6:30 p.m.
 Mon., May. 14:
 Office Hours: 3:30  4:30 p.m.
 Fri., May. 11:
 Office Hours: 2:30  3:30 p.m.
Assignments are listed by the date due.
If your browser is prepared to handle
MathML, you should use the XHTML
version of this page; if not, you should view the
classical HTML version.
PDF and DVI
(requires TeX software) versions
of this page are available for printing.
Most of these assignments are simply casual exercises designed to
prepare you for tests and the written assignments. While you may find
it helpful to discuss the casual exercises with others,
no collaboration is permitted on the written assignments.
 Mon., May. 7:
 Written Assignment No. 5 (PDF for printing  classical HTML for terminal window browsing) is due.
 Wed., May. 2:
 Exercises:
Let E denote the cubic curve y^{2} + x y + 2 y = x^{3} + 3.
Find a point on E over the field Z/5Z of largest
possible order.
Can every one of the printable ASCII codes (range
32  127) actually be represented by a point of E in the field
Z/1283Z by the method described in class (and on the next
assignment sheet) when up to 10 attempts are made for each value?
If not, which characters cannot be represented? What happens if
11 attempts are made for each value?
Continuing in the vein of the previous exercise represent the
3 character string “abc” as a point on E in the field
Z/1283Z with up to 10 attempts at representation for each
value and then encrypt the resulting sequence of 3 points on E
using the public key
b = [147, 376] c = [706, 905] .

What secret key was used to construct the public key of the
previous exercise?
 Mon., Apr. 30:
 Exercises from the text:
 262: 6, 7, 12  14, 16, 17, 20, 21, 24, 25
 Wed., Apr. 25:
 Read: §§ 10.5, 10.6
 224: 3  5
 260: 3  5
 Mon., Apr. 23:
 Written Assignment No. 4 (PDF for printing  classical HTML for terminal window browsing) is due. (The due
date had previously been set as Wednesday, April 18.)
Code for addition of points on cubic curves will be added to
the web at http://www.albany.edu/~hammond/maple/
after this class meeting.
 Wed., April. 18:
 Study the slides (also available as PDF or DVI or classical HTML) about addition of points on
cubic curves.
The slides now incorporate material from our class on April 18 as
well as the class on April 16. They were last revised on April 19.
 Mon., Apr. 16:
 Read: §§ 10.1  10.2
 260: 1, 2
 And this:
Find the area enclosed by the loop of the cubic curve y^{2} = x^{3}  x.
Repeat for the cubic curve y^{2} = x  x^{3}.
 Wed., Apr. 11:
 Read: §§ 9.1  9.2
 224: 1, 2
 And this:
Find the polynomials p_{n}(x) such that
{d^{n}}/{d x^{n}} exp(1/x^{2}) = {p_{n}(x)}/{x^{3n}} · exp(1/x^{2})

for 1 <= n <= 7. Can you give a general recursive formula
for p_{n}(x)?
 Apr 2  9
 University Recess: no classes
 Wed., Mar. 28:
 Written Assignment No. 3 (PDF for printing  classical HTML for terminal window browsing) is due.
 Mon., Mar. 26:
 Read: §§ 7.4  7.6, 8.6  8.8
Do these: 188: 3, 4, 5
Explore the Maple function for finding primitive roots mod m,
which is numtheory[primroot].
 a. Find the smallest primitive root modulo 289 that
is larger than 100.
 b. Find the smallest positive nonprime primitive root
mod 40487.
 c. Find the smallest positive number that is primitive
modulo both 101 and 103. Is it primitive mod 101*103?
 d. If c is primitive modulo both 101 and 103, what
congruence condition on integers j, k >= 0 is equivalent to the
condition that c^{j} EQUIV c^{k} (mod 101*103)?
Let p be the prime 128^{15} + 39. Without trying to solve determine
which of the following two congruence equations is solvable:
2^{m}EQUIV 11 (mod p) ਊnd 11^{n}EQUIV 2 (mod p) .

Are you able to solve the solvable one?
 Wed., Mar. 21:
 Read: §§ 7.1  7.3
 174: 5
 188: 1, 2
 And this:
Continuing in the context of the last exercise in the previous
assignment, you are now being told that the squeezed vector

[712147006187606979338143444233878549915653153140991743218564586,

1786621100356707079804781015651798041041290004401049203827247506,

1782184643903441535885937756067735301974983951149305281678962346,

1639000008839632707546680167815675641387259213687418193657940006,

1535960089185549654706004534787094483505037489361312984436350635,

1195799297844909964188410557114692983427064185633447219054911622,

1529236902471918734371483225353942522875473990416411009757742702,

409979669999633360347425246927425729369778446996539051720679885,

1805600608974788719838347443426498779266916648865325622675849897,

1058983644708927766918309320955981103594250701210512127725439642]
(where k is maximum, as before, for the given modulus m) may be
decrypted with the exponent
d = 679417638057246102387290084428241348920601574129013039486178441 .

 A. Decrypt it, expand its terms in base 128, and convert the
resulting vector, regarded as a sequence of ASCII
codes, to a string.
 B. Can you determine what the encrypting exponent was?
 Mon., Mar. 19:
 Read: §§ 6.1  6.3
 174: 1  3; disregard the last sentence in exercise 1.
 And this: Given a vector of digits in base 128 what is the
largest block size k for squeezing the vector into a vector of digits
for base 128^{k} so that the resulting squeezed vector can be faithfully
encrypted by taking a suitable power of each entry modulo the integer
m = 2468256835981809063232453773840873253369376547681693188080273739

under the hypothesis, which is satisfied here, that the integer m
is squarefree?
 Wed., Mar. 14:
 Midterm Test in class
 Mon., Mar. 12:
 A light assignment prior to the midterm test:
 • Bring review questions.
 • Use network resources to find Maple code for solving
Sudoku puzzles, and then find out how long it takes Maple
to solve this one:
  8     1   2   7   1     9  4     3    5     8    4   9       8  1  5       1   6    4     5    7     7  9     6   1   6   4     2   

WARNING: Be careful when downloading code from the network.
First make sure a location where you find code is trustworthy,
and then look over any code before running it.
 Wed., Mar. 7:
 Written Assignment No. 2 (PDF for printing  classical HTML for terminal window browsing) is due.
Code for vector shifting of the type used in problem 5 may be
found at http://www.albany.edu/~hammond/maple/.
 Mon., Mar. 5:
 Read: §§ 5.3  5.4
 151: 3  5
 And this: Conduct some experiments in
cryptography using computers (PDF for printing  classical HTML for terminal window browsing).
 Wed., Feb. 28:
 Announcement: The midterm test will be held on Wednesday, March 14.
Read: §§ 5.1  5.2
 151: 1, 2
 And this: Write a Maple procedure that given a univariate polynomial
f(x) and a polynomial b(x) of degree at least 1 returns the vector
of coefficients c_{j}(x) for the badic expansion of f(x)
f(x) = SUM_{j >= 0}^{j}[ c_{j}(x) b(x) ]

where deg(c_{j}(x)) < deg(b(x)) for each j >= 0.
 Mon., Feb. 26:
 Scan: Chapter 4
Exercises:
 137: 1, 4
 And this:
Write a Maple procedure that given a base b >= 2 and a triple of
vectors equivalent to the base b representation of a positive
rational number  each vector consisting of digits relative to the
base b, with the vectors in order being (a) the digit sequence
(possibly empty) to the left of the decimal point, (b) the digit
sequence (possibly empty) to the right of the decimal point before the
repetition pattern, and (c) the digit sequence (if any) that repeats
 returns the positive rational number as a fraction m/n where m
and n are positive integers without common divisor.
 Week: Feb 19  23
 University Recess: no classes
 Wed., Feb. 14:
 No meeting.
The University has announced that, due to severe snow conditions,
all day and evening classes on February 14 are cancelled.
Comment on exercise 93: 6 (PDF for printing  classical HTML for terminal window browsing)
 Mon., Feb. 12:
 Read: §§ 3.4  3.6
Exercises:
 93: 6  10
 And this:
Write a Maple procedure that when given a finite continued fraction,
presented as the vector
[  a_{0},a_{1},a_{2},a_{3}, …, a_{n}  ] 
representing
a_{0} + {1}/{a_{1}+{1}/{a_{2} + {1}/{… + a_{n}}}} ,

with the a_{i} all integers and a_{i} >= 1 for i >= 1, returns
the rational number it represents.
 Wed., Feb. 7:
 Written Assignment No. 1 (PDF for printing  classical HTML for terminal window browsing) is due.
 Mon., Feb. 5:
 Read: §§ 3.1  3.3
Exercises:
 63: 12, 13
 93: 1  5
 And this: Examine all iterates of the Syracuse function
applied to each integer n up to 10,000 and find the integer n
in that range having an iterate s_{k}(n) for which the ratio s_{k}(n)/n
of the iterate to the starting integer is largest. Hint: If
the problem is modified to consider only integers n up to 100, then
the integer in that smaller range having an iterate with largest ratio
is 27, and the iterate presenting the largest ratio is s_{77}(27) = 9232.
 Wed., Jan. 31:
 Read: §§ 2.5  2.6
Exercises:
 63: 6  11
 And this: ssq will be the name for the function defined by
ssq(n, b) = 1 +  (  sum of the squares of the base b digits of n  ) 
 .

Maple code for this function may be found at
http://www.albany.edu/~hammond/maple/. In that code
if the second variable b is not specified, then it is understood
to be 10.
Conduct experiments with the base b having the values 2, 3, 5,
and 6 to try to determine what happens when ssq is iterated
starting from various positive integers n.
 Mon., Jan. 29:
 Read: §§ 2.1  2.4
Exercises:
 63: 1  5
 And this: The Syracuse function s is defined for integers n by
s(n) =  {  
1 
if n <= 1 
3n + 1 
if n > 1 is odd 
n/2 
if n > 1 is even 
 


The iterates of s are
s_{1}(n) = s(n), s_{2}(n) = s(s(n)), s_{3}(n) = s(s(s(n))),… .

For example, s_{1}(6) = s(6) = 3, s_{2}(6) = s(3) = 10, s_{3}(6) = s(10) = 5,
s_{4}(6) = s(5) = 16, s_{5}(6) = s(16) = 8, s_{6}(6) = s(8) = 4,
s_{7}(6) = s(4) = 2, s_{8}(6) = s(2) = 1. Since the 8th iterate of s
applied to 6 is 1, all higher iterates of s applied to 6
are 1.
Find the 5 smallest values of n for which the first 2n + 1
iterations of s applied to n fail to yield 1.
 Wed., Jan. 24:
 Acquire the textbook.
Read through chapter 1, and try some of
what is sketched there for yourself in Maple.
About free general purpose computer algebra systems: The
following items were found through a web search, but none of them have
been reviewed.
 Axiom

Axiom has been in development since 1973 and was
sold as a commercial product. It has been released as free software
under the Modified BSD License. It is sponsored by CAISS, the Center
for Algorithms and Interactive Scientific Software, at The City
College of New York.
 Maxima

Maxima is a descendant of Macsyma, the computer
algebra system developed in the late 1960s at the Massachusetts
Institute of Technology. It is free under the GNU General Public
License subject to some export restrictions from the U.S. Department
of Energy. A proprietary version of Macsyma is also
available.
 Mon., Jan. 22:
 First meeting: No assignment.
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