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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 9 "Lecture 2" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 40 "Shift + Return key versus the Return key
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 266 
78 "If the cursor is on the same line as a well formed Maple command, \+
hitting the " }{TEXT 268 6 "Return" }{TEXT 269 44 " key will cause the
 command to be executed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 267 246 "Sometimes commands are several lines long and yo
u don't want Maple to execute the commands or sequence of commands til
l you have typed in all the lines. To do this one needs to know how to
 go to another output line without getting a new prompt, t" }{TEXT -1 
25 "hat is, how to suppress  " }{TEXT 256 5 ">.   " }{TEXT 270 21 "In \+
such cases use of " }{TEXT 257 14 "Shift + Return" }{TEXT -1 16 " (hol
d down the " }{TEXT 258 5 "Shift" }{TEXT -1 17 " key and then hit" }
{TEXT 259 7 " Return" }{TEXT -1 63 ") will accomplish this. \n\nIf the
 above doesn't work look under " }{TEXT 261 4 "Edit" }{TEXT -1 16 " an
d go down to " }{TEXT 260 13 "Preferences. " }{TEXT -1 10 "Make sure \+
" }{TEXT 262 29 "Execute Input with Return Key" }{TEXT -1 24 " is chec
ked.  Otherwise " }{TEXT 264 6 "Return" }{TEXT -1 74 " will move one t
o the next line  and  execution will occur  only with the " }{TEXT 
263 5 "Enter" }{TEXT -1 5 " key." }{TEXT 265 0 "" }{TEXT -1 105 "\n\nB
efore continuing make sure you can type in the following without getti
ng more that the initial prompt " }{TEXT 271 1 ">" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "for i from 1 to 3 do \n  pri
nt(\"Hello\");\nend do;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "for . . .  do . . .  end d
o" }}{EXCHG {PARA 0 "" 0 "" {TEXT 276 5 "Loops" }{TEXT -1 4 " or " }
{TEXT 277 8 "do loops" }{TEXT -1 126 " provide a way to repeat operati
ons a specified number of times. They are best understood by examples.
  Execute the following:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i from 1 to 5 do \n  i,i
^2,i^3;\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 278 "Now we illus
trate a common technique for adding a specified number of terms. There
 is nothing special about SUM as a variable. We could use any name for
 this variable. We will find the sum of the integers 1 through 10.    \+
 (Note that we start by setting the variable SUM to 0.) " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "SUM:=0;\n" }{TEXT -1 0 "" }{MPLTEXT 
1 0 46 "for i from 1 to 10 do \n SUM:=SUM + i; \nend do;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 53 "Note the value of i after the loop has be
en executed:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "i;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The value of SUM after the loop ha
s been executed:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "SUM;" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "To suppress the intermediate step
s we may do place a colon after " }{TEXT 322 6 "end do" }{TEXT -1 35 "
 instead of a semi-colon and place " }{TEXT 324 4 "SUM;" }{TEXT -1 7 "
 after " }{TEXT 323 8 "end do: " }{TEXT -1 40 "This way we just get th
e result desired." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "SUM:=0
:\nfor i from 1 to 10 do \n SUM:= SUM + i; \nend do:\nSUM;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "For another example we compute " }
{XPPEDIT 18 0 "Sum(1/k,k = 1 .. 100);" "6#-%$SumG6$*&\"\"\"F'%\"kG!\"
\"/F(;F'\"$+\"" }{TEXT -1 32 ": This time we use the variable " }
{TEXT 316 5 "total" }{TEXT -1 60 " to keep track of the sum. Of course
, any variable would do." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 
"total:=0:\nfor k from 1 to 100 do \n total:=total+1/k; \nend do:" }
{TEXT -1 0 "" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "total;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e
valf(total);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?modp" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "if . . . then . . . end if" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 303 31 "B
efore giving an example of an " }{TEXT 308 22 "if ... then ... end if
" }{TEXT 309 26 " statement we discuss the " }{TEXT 311 3 "mod" }
{TEXT 312 23 " procedure. The command" }{TEXT 307 9 " n mod d " }
{TEXT 325 15 "or equivalently" }{TEXT 326 11 " modp(n,d) " }{TEXT -1 
67 "gives the remainder when the integer n is divided by the integer d
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "12 mod 10, modp(12,10)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "12 mod 2, modp(12,2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "12 mod 7, modp(12,7);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Note that if " }{TEXT 306 11 "n
 mod d = 0" }{TEXT -1 13 " then d is a " }{TEXT 304 6 "factor" }{TEXT 
-1 4 " or " }{TEXT 305 7 "divisor" }{TEXT -1 102 " of n. Let's find al
l divisors of 126; Notice the way we indent here, so it is easier to i
dentify the " }{MPLTEXT 1 0 0 "" }{TEXT 313 9 "do....end" }{TEXT -1 1 
" " }{TEXT 314 2 "do" }{TEXT -1 6 "  and " }{TEXT 315 13 "if ... end i
f" }{TEXT -1 15 " constructions." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 85 "n:=126;\nfor d from 1 to n do \n  if n mod d = 0 then
 \n    print(d); \n  end if;\nend do:" }{TEXT -1 0 "" }{MPLTEXT 1 0 1 
"\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Change the value of n abov
e and re-execute the procedure. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 33 "Here's another example using the " }
{TEXT 310 22 "if ... then ... end if" }{TEXT -1 14 " construction." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "for k from 1 to 50 do\n  a
:=3*k^2 + 10*k + 1;\n  r:= modp(a,31);\n  if r  <  3 then  \n    print
(k,r); \n  end if;\nend do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "No
te that the above could be accomplished as follows:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 99 "for k from 1 to 50 do\n  r:=modp(3*k^2 + \+
10*k + 1,31);\n  if r  <  3 then print(k,r); end if;\nend do:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 42 "The Set of Positive Divisors of an Integer" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Now we show
 how to form the " }{TEXT 273 3 "set" }{TEXT -1 118 " of all positive \+
divisors of 126.\nA student of number theory will easily see how to im
prove this little program.  The " }{TEXT 317 10 "printlevel" }{TEXT 
-1 75 " is 1 by default. Changing it to 2 allows us to see what's happ
ening below:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "printlevel:
=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "n:=126:\nS:=\{\}; " 
}{TEXT -1 49 "Start with the empty set which is denoted by \{ \}." }
{MPLTEXT 1 0 83 "\nfor d from 1 to n do\n if n mod d = 0 then \n   S:=
 S union \{d\}; \n end if;\nend do;\n\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 39 "Let's see what values d and S have now." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "d,S;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 9 "Note the " }{TEXT 272 22 "if .. then ... end if;" }{TEXT -1 15 "
 construction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 261 "Now move the cursor back to printlevel:=2; , change the \+
2 to a 1 and re-execute all the above commands. Note that the intermed
iate steps are now suppressed. Note that you may also change the assig
nment n:=126; to n:=1284; or anything you please and re-execute. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 61 "A simpler way to handle the problem in general \+
is to build a " }{TEXT 318 9 "procedure" }{TEXT -1 89 " to find the se
t of all positive divisors  of any integer n. We will call this proced
ure " }{TEXT 319 8 "Divisors" }{TEXT -1 131 ". Examine this procedure \+
closely. Note the use of return on the next to last line of the defini
tion of this procedure.  DO NOT USE " }{TEXT 327 5 "print" }{TEXT -1 
13 " in place of " }{TEXT 328 6 "return" }{TEXT -1 40 ".  We will expl
ain the difference later." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "Divisors:=proc(n)\nS:=\{\}:\nfor d
 from 1 to n do\n if n mod d = 0 then \n   S:=S union \{d\}; \n end if
;\nend do;\nreturn S;\nend proc;\n" }}}{EXCHG {PARA 258 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 
239 "Notice the warning messages when you execute the statement defini
ng the above procedure. Also notice that Maple reprints the procedure \+
if you end the definition with ;  . We remedy both of these problems i
n the following corrected version." }{TEXT 280 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "Divisors:=
proc(n)\nlocal S,d;\nS:=\{\}:\nfor d from 1 to n do\n if n mod d = 0 t
hen S:=S union \{d\}; end if;\nend do;\nreturn S;\nend proc:\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 278 10 "Important:" }{TEXT 320 174 " Make
 it a habit to end a procedure with : instead of  ; to save output--es
pecially when printing your worksheet. Also ALWAYS declare all variabl
es used in a procedure to be " }{TEXT 329 5 "local" }{TEXT 330 61 " as
 we do above, unless there is a good reason not to do so. " }{TEXT -1 
45 "Later we will discuss the difference between " }{TEXT 274 5 "local
" }{TEXT -1 5 " and " }{TEXT 275 6 "global" }{TEXT -1 41 " variables.
\n\nNow lets test the procedure " }{TEXT 279 8 "Divisors" }{TEXT -1 
50 ": Try somethings first that we know the answer to:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Divisors(1);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "Divisors(2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "Divisors(6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "Divisors(126);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Div
isors(12345);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "More Examples of Procedures " }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Let's look again at how to compute
 a sum such as " }{XPPEDIT 18 0 "Sum(i^2,i = 1 .. 4);" "6#-%$SumG6$*$%
\"iG\"\"#/F';\"\"\"\"\"%" }{TEXT -1 85 ". Let's try to make it general
 by introducing a variable n and setting it equal to 4:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "n:=4:\nTotal:=0:\nfor i from 1 to n
 do\n  Total:=Total+i^2:\nend do:\nTotal;\n" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 152 "We could simply change the assignment of n:=4 to n:=10
 or any other value and re-execute to find other sums. Instead, let's \+
make a procedure to compute " }{XPPEDIT 18 0 "Sum(i^2,i = 1 .. n);" "6
#-%$SumG6$*$%\"iG\"\"#/F';\"\"\"%\"nG" }{TEXT -1 65 " for any given po
sitive integer n. We will call this procedure F:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 121 "F:=proc(n)\n  local i,Total;\n  Total:=0:\n \+
 for i from 1 to n do\n    Total:=Total+i^2:\n  end do;\n  return Tota
l;\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "F(4);\n" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F(1000);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 38 "Here's  a procedure that will compute " }
{XPPEDIT 18 0 "sum(f(i),i = a .. b);" "6#-%$sumG6$-%\"fG6#%\"iG/F);%\"
aG%\"bG" }{TEXT -1 135 " where a and b are any integers and f is any f
unction. We will call the procedure S. Note that it requires three arg
uments: f, a and b:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "S:=
proc(f,a,b)\nlocal i, Total;\nTotal:=0;\nfor i from a to b do\n  Total
:=Total + f(i);\nend do;\nreturn Total;\nend proc:\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 37 "Some applications of our procedure S:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "S(sin,1,5);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "S(x->x^2, 1, 6);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "S(f,-3,5);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Here is a procedure name
d H which has the property that H(n) = big if  n is greater than 100 a
nd H(n) =  little if not." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "H:=proc(n)\nif n > 100 then \n  ret
urn big; \nfi;\nreturn little;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 220 "Note that if n > 100 is true t
hen H immediately returns the value big and therefore does not proceed
 to the next statement. On the other hand, if n > 100 is false then Ma
ple skips down to the last line a returns little. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "H(221);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "H(11);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Cutt
ing and Pasting" }}{PARA 0 "" 0 "" {TEXT -1 134 "When constructing a p
rocedure it is a good idea to test the commands individually first so \+
you can be sure what is happening. One may " }{TEXT 331 3 "cut" }
{TEXT -1 2 ", " }{TEXT 332 4 "copy" }{TEXT -1 5 " and " }{TEXT 333 5 "
paste" }{TEXT -1 250 " text and Maple input and output pretty much at \+
will.  This is useful when constructing a procedure from some program \+
that you have already tested. You may illustrate this by copying the p
rogram below and making it into a procedure as we did above.  " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "n:=4:\nTotal:=0:\nfor i from
 1 to n do\n  Total:=Total+i^2:\nend do:\nTotal;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 "Order of Execution of Comma
nds in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 347 "In a single execution group Maple executes the command
s in order from top to bottom. However, if you go back and change some
 computation in a different execution group then it does not change th
e following results unless the entire worksheet is executed once more.
 Here is a simple example. First execute the following commands in the
 order given." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=2;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y:=3;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 7 "z:=x*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 2 "z;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "Now go back and cha
nge the 2 in the line x:=2 to a 5 and execute that command. Notice tha
t the value of z has not changed.  To change the value of z to 15 we m
ust re-execute the line z:=x*y." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 2 "z;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 224 "Of course, we would
 not have this problem if we put all the three commands in a single ex
ecution groups as follows: But after changing the 2 to a 5 in x:=2; we
 must execute the execution group for the changes to take effect." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x:=2:\ny:=3:\nz:=x*y:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "z;" }}}{EXCHG {PARA 258 "" 0 
"" {TEXT -1 176 "The moral of this is that when executing the lines in
 a Maple worksheet it is important which order they are executed in ra
ther than the order in which they appear on the page." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 40 "Assignment 2 (Due Wednesday, January 23)" }}{EXCHG {PARA 0 "" 
0 "" {TEXT 295 334 "From now on:  25% will be deducted from each assig
nment which does not conform to the format described in Assignment 1 P
roblem 1 (a).  Also some points will be deleted for any procedure that
 use local variables that are not declared to be local in the body of \+
the procedure and any procedure  definition that does not end in a col
on. " }{TEXT 287 12 "\n\nProblem 1." }{TEXT -1 31 "  \n(a) Write a pro
gram using a " }{TEXT 321 7 "do loop" }{TEXT -1 20 " to compute the su
m " }{XPPEDIT 18 0 "sum((2/3)^n,n = 2 .. 100);" "6#-%$sumG6$)*&\"\"#\"
\"\"\"\"$!\"\"%\"nG/F,;F(\"$+\"" }{TEXT -1 9 ".  Apply " }{TEXT 288 5 
"evalf" }{TEXT -1 94 " to obtain a floating point approximation to the
 sum.\n\n(b) Write a program that will find the " }{TEXT 289 3 "set" }
{TEXT -1 107 " S of all positive integers less than 1000 that have 11 \+
as a factor.  [n has 11 as a factor if and only if " }{XPPEDIT 18 0 "`
mod`(n,11) = 0;" "6#/-%$modG6$%\"nG\"#6\"\"!" }{TEXT -1 2 ".]" }}
{PARA 0 "" 0 "" {TEXT 284 12 "\nProblem 2. " }{TEXT -1 20 "\n\nWrite a
 procedure " }{TEXT 296 7 "ISPRIME" }{TEXT -1 57 " whose input is a po
sitive integer n and whose output is " }{TEXT 301 4 "true" }{TEXT -1 
19 " if n is prime and " }{TEXT 302 5 "false" }{TEXT -1 58 " otherwise
.  Test the procedure on the following numbers: " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "1,  2, 3, 4, 111, 531, 79
19, " }{XPPEDIT 18 0 "15689;" "6#\"&*o:" }{TEXT -1 15 ", and 262681.  \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The a
nswers should be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "false,true,true,false,false,false,true,
false,true;" "6+%&falseG%%trueGF$F#F#F#F$F#F$" }{TEXT 291 9 ". \n\nHin
t:" }{TEXT -1 36 " Here's a start for the program:\n\n> " }{MPLTEXT 1 
0 111 " ISPRIME:=proc(n)\n   local d;\n   if n < 2 then return false; \+
end if;\n   if n = 2 then return true; end if;\n   \n" }{TEXT -1 47 "N
ote that if n < 2 then the third  line causes " }{TEXT 297 10 "ISPRIME
(n)" }{TEXT -1 55 " to return false. If n = 2 then the fourth line cau
ses " }{TEXT 298 7 "ISPRIME" }{TEXT -1 424 " to return true. If neithe
r is true then n is at least 3.  So now you only need to handle in the
 rest of the program the case that n is at least 3. This can be done b
y using a loop to check whether or not n is divisible by an integer d \+
from 2 to n-1. If you find such a value of d you want to return false.
 If the loop is completed without finding any such d then you should r
eturn true. You may be able to improve on this.\n" }}{PARA 0 "" 0 "" 
{TEXT 290 6 "Remark" }{TEXT -1 48 ". Maple actually has a built-in pro
cedure named " }{TEXT 294 7 "isprime" }{TEXT -1 47 " which performs th
e same task as the procedure " }{TEXT 299 7 "ISPRIME" }{TEXT -1 16 " i
n Problem 2.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 18 "Maple's procedure " }{TEXT 337 7 "isprime" }{TEXT -1 13 "
 is really a " }{TEXT 339 31 "probabilistic primality tester." }{TEXT 
-1 5 " See " }{TEXT 338 8 "?isprime" }{TEXT -1 28 ".  To see how the p
rocedure " }{TEXT 336 7 "isprime" }{TEXT -1 36 " is constructed execut
e the command " }{TEXT 335 19 "showstat(isprime); " }{TEXT -1 197 "How
ever, it is not easy to see what the program is doing. So I don't reco
mmend spending a lot of time on trying to figure it out unless you hav
e lots of experience in programming and number theory." }}{PARA 0 "" 
0 "" {TEXT -1 1 "\n" }{TEXT 285 10 "Problem 3." }{TEXT -1 60 " \n\nIf \+
n is a positive integer greater than 2 that satisfies " }{XPPEDIT 18 
0 "modp(2^(n-1),n) = 1;" "6#/-%%modpG6$)\"\"#,&%\"nG\"\"\"F+!\"\"F*F+
" }{TEXT -1 39 " and n is not prime then n is called a " }{TEXT 334 
13 "2-pseudoprime" }{TEXT -1 95 ". Find all 2-pseudoprimes less than 2
000 and give the factorization of each one that you find. " }{TEXT 
340 4 "Hint" }{TEXT -1 44 ": The smallest 2-pseudoprime is 341 = 11x31
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT 286 11 "\nProblem 4." }{TEXT -1 27 "  \n\n(a) Wr
ite a procedure, " }{TEXT 281 8 "primesum" }{TEXT -1 92 ", whose input
 is a positive integer n and whose output is the sum of all primes p s
uch that " }{XPPEDIT 18 0 "p <= n;" "6#1%\"pG%\"nG" }{TEXT -1 37 ". Yo
u may use either your procedure  " }{TEXT 300 7 "ISPRIME" }{TEXT -1 
42 " from problem 2 or the built-in procedure " }{TEXT 282 9 "isprime.
 " }{TEXT -1 46 "Test the procedure for n = 2, 4, 10, and 1000." }}
{PARA 0 "" 0 "" {TEXT -1 25 "\n (b) Write a procedure, " }{TEXT 292 8 
"primeset" }{TEXT -1 93 ", whose input is  a positive integer n and wh
ose output is the set of all primes p such that " }{XPPEDIT 18 0 "p <=
 n;" "6#1%\"pG%\"nG" }{TEXT -1 87 ". You may use either your procedure
  ISPRIME from problem 2  or the built-in procedure " }{TEXT 293 9 "is
prime. " }{TEXT -1 46 "Test the procedure for n = 2, 4, 10, and 1000.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 1 " " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }
}}}{MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 
33 1 1 }