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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 39 "Introduction to Maple for
 Number Theory" }}}{EXCHG {PARA 19 "" 0 "" {TEXT -1 57 "Mike O'Sulliva
n\nSan Diego State University\nSeptember 2004" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 
"" {TEXT -1 25 "Sets, Lists, some basics." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "A set is denoted by \{\} an
d a list by [].  Both may be empty." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S1:= \{\};  S2 := \{1, 23
\}; S3 := \{23,4,5\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "Union, \+
intersection and minus (set difference) can be accomplished two ways. \+
 Notice the back quote.  The second form is more compact for more than
 two sets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 49 "T1 := S2 union S3;  T2 := `intersect`(S2, S3,T1);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "You can test for membership and
 the result is of type  boolean." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "b := member(1, S1);  c := member(1, S2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "type(b,boolean);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "b and c;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Lists
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "L1 := [2,3,5,2,3,6,2,3]
; L2 := [3,4,8,9];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "nops(
L1); op(3,L1);  [op(1..6,L1)]; \{op(1..6,L1)\};" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "convert(L1,set);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "L3 := [op(L1),op(L2)];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "L3 := map(x -> x^2, L2);" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Divisibili
ty" }}{PARA 0 "" 0 "" {TEXT -1 73 "Maple has two functions to compute \+
quotients and remainders of integers, " }}{PARA 0 "" 0 "" {TEXT -1 18 
"iquo() and irem()." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "q := \+
iquo(10, 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "r := irem(1
0,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "q*3 + r;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "You can also use iquo() to get the
 remainder (and irem() to get the quotient)." }}{PARA 0 "" 0 "" {TEXT 
-1 17 "Notice the quotes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "r := irem(29,7,'q');  q
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "q*7 + r;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 72 "The floor of a quotient of two integers i
s the iquo of the two integers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "floor(29/7);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 87 "While we're discussing divisibility, try the fu
nction ifactor().  You might think about" }}{PARA 0 "" 0 "" {TEXT -1 
43 "what this output is and why it makes sense." }}{PARA 0 "" 0 "" 
{TEXT -1 62 "Note: The output of ifactors() is  in the form [prime, po
wer]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "ifactor(-60);ifactors(-60);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 182 "Using Maple Help you can search for \"divisors\".  It \+
is one of the functions in the number theory package.  You might want \+
to browse that package to see the other functions available." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n
umtheory[divisors](30);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "Divisibility
 for Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 91 "The functions used fo
r quotient and remainder of polynomials are similar to those used for \+
" }}{PARA 0 "" 0 "" {TEXT -1 47 "integers, but you need to specify the
 variable." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := x^5+x+1; \+
 g := x^2+1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "quo(f,g, x)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Now try the remainder  funct
ion rem(), and check that the result you get is correct." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 34 "Primes and the Euclidean Algorithm" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "There are several \+
ways to get primes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "next
prime(100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p := ithprim
e(55);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "You can test for primal
ity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "isprime(p);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "The Euclidean algeorithm is used \+
to find the greatest common divisor.  You can get the quotient of each
 integer by the gcd.    The two extra variables are set to these value
s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "gcd(27,18);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gcd(27,18,'ap','bp');ap; bp;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "You can also get the linear c
ombination that gives the gcd.." }}{PARA 0 "" 0 "" {TEXT -1 64 "Note \+
\"igcdex\"  is the extended Euclidean algorithm for integers." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "igcdex(45,290,'s','t');" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s; t; s*45 + t*290;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "The same can be done with polynomi
als.  You have to specify the variable, too." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "F := x^3-3*x-2; H
 := x^4-4*x^2-x+2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gcdex
(F, H, x, 's','t');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s; t
;  s*F + t*H;  simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Try using the functions ilcm() a
nd  lcm()." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 49 "Iterative and conditional statements (For
 and If)" }}{PARA 0 "" 0 "" {TEXT -1 82 "Two of the most useful things
 in programming are iterative loops and conditionals." }}{PARA 0 "" 0 
"" {TEXT -1 30 "Here are a couple of examples." }}{PARA 0 "" 0 "" 
{TEXT -1 57 "The syntax can be fussy, so use indetation to keep track.
" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here are the basic templates.  You ma
y have one or more statements inside the " }}{PARA 0 "" 0 "" {TEXT -1 
56 "for loop (or then clause).  Separate them by semicolons." }}{PARA 
0 "" 0 "" {TEXT -1 22 "for i from __ to __ do" }}{PARA 0 "" 0 "" 
{TEXT -1 13 "     _______;" }}{PARA 0 "" 0 "" {TEXT -1 12 "    _______
;" }}{PARA 0 "" 0 "" {TEXT -1 7 "end do;" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "if ___ then " }}{PARA 0 "" 0 "" 
{TEXT -1 9 "     ___;" }}{PARA 0 "" 0 "" {TEXT -1 8 "    ___;" }}
{PARA 0 "" 0 "" {TEXT -1 7 "end if;" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "for i from 1 to 10 do\n  i^2;\nend do;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "16;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "for i from 1 to 100 do\n   if isprime(i) then print(i
);\n   end if;\nend do;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "  Exe
rcises" }}{PARA 0 "" 0 "" {TEXT -1 53 "1.   Make a list with the first
 20 Fibonacci numbers." }}{PARA 0 "" 0 "" {TEXT -1 82 "      Next, giv
en a and b, and starting values g_0 and g_1, produce the sequence  " }
}{PARA 0 "" 0 "" {TEXT -1 48 "      determined by g_n = a g_\{n-1\} + \+
b g_\{n-2\}." }}{PARA 0 "" 0 "" {TEXT -1 212 "2.  Make a list of all t
he Fibonacci numbers (less than some bound) which are also perfect squ
ares.  List also the index of the Fibonacci number.  Try it by a brute
 force method and then try to be more efficient." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "3.  Which Fibonacci numbe
rs are  divisible by 7, by 8, etc. " }}{PARA 0 "" 0 "" {TEXT -1 31 "  \+
   Can you form a conjecture?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
256 0 "" }{TEXT 257 0 "" }{TEXT -1 54 "4.  For several real values of \+
 x compute  [x] + [-x] " }}}}{MARK "1 0 0" 57 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }