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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 18 "Maple Worksheet #3" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Random Numbers" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 21 "Here we use the term " }{TEXT 259 13 "random numb
er" }{TEXT -1 70 ", but actually the numbers we shall generate are mor
e properly called " }{TEXT 260 21 "pseudo-random numbers" }{TEXT -1 
151 ". Since they are generated by a computer they are not random in t
he true sense of the word. To read more about this subject see, for ex
ample, the book " }{TEXT 257 40 "The Art of Computer Programming Vol. \+
2, " }{TEXT 258 9 "Chapter 2" }{TEXT -1 18 ", by Donald Knuth." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Th
e smallest n digit positive integer is " }{XPPEDIT 18 0 "10^(n-1);" "6
#)\"#5,&%\"nG\"\"\"F'!\"\"" }{TEXT -1 45 " and the largest n digit pos
itive integer is " }{XPPEDIT 18 0 "10^n-1;" "6#,&)\"#5%\"nG\"\"\"F'!\"
\"" }{TEXT -1 25 ". For example, for n = 3:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 12 "10^3;10^4-1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
11 "For n = 40:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "10^39;10
^40 - 1;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Using " }{TEXT 267 6 "
length" }{TEXT -1 68 "(x) we can obtain the number of digits in the nu
mber x. For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "2^5
0;\nlength(2^50);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "To create a \+
procedure, call  it " }{TEXT 256 4 "roll" }{TEXT -1 80 ", that will ge
nerate a random number (integer) between a and b use the command: " }
{TEXT 296 17 "roll:=rand(a..b);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "We could call the procedure anythi
ng we want. I just call it " }{TEXT 295 4 "roll" }{TEXT -1 32 " to sug
gest a roll of the dice. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "For \+
example " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "roll:=rand(0..9
):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "This procedure is used with
 just () as input:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "roll()
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "roll();" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for i from 1 to 4 do\nprint(roll())
;\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "To make a procedure
 to generate random 40 digit numbers we write:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "roll_40:=rand(10^39..10^40 - 1):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "for i from 1 to 5 do\nroll_40();\ne
nd do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Of course one could rep
lace 40 by any positive integer." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "Random n-Di
git Primes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 72 "We may use the following method to generate two random 10
0-digit primes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "roll:=ra
nd(10^99..10^100-1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "p:=
nextprime(roll());\n\nq:=nextprime(roll());" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 40 "Now we generate random 200 digit primes:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "roll:=rand(10^199..10^200-1):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p:=nextprime(roll());" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "q:=nextprime(roll());" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Notice that it takes a while to g
enerate these. Maple first generates a random 200 digit number and the
n " }{TEXT 262 9 "nextprime" }{TEXT -1 113 " searches through all numb
ers greater than that number, one at a time, till a prime is found usi
ng the procedure " }{TEXT 261 7 "isprime" }{TEXT -1 1 "." }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 25 "Timing Maple Computations" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 12 "The command " }{TEXT 294 6 "time()" }
{TEXT -1 146 " returns the total CPU time used since the start of the \+
Maple session. The units are in seconds and the value returned is a fl
oating-point number." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "t1:
=time();" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "t2:=time();" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Here's how one may time a comput
ation. In this case we time how long it takes to find the next prime f
ollowing " }{XPPEDIT 18 0 "10^100;" "6#*$\"#5\"$+\"" }{TEXT -1 1 "." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "starting_time:=time();\nne
xtprime(10^100);\ntime_required:=time() - starting_time;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 109 "Let's factor as many Mersenne primes as \+
we can find in 1 second. We will stop when we have exceeded 1 second.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "starting_time:=time():
\nfor n from 1 to 100 do\nif time() - starting_time > 1 then break; en
d if:\np:=ithprime(n):\nq:=2^p-1:\nif isprime(q) then print(p); end if
;\nend do:\nelapsed_time:=time() - starting_time;\n" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Computing powers mod m" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The maple command for computing " 
}{XPPEDIT 18 0 "`mod`(a^n,m);" "6#-%$modG6$)%\"aG%\"nG%\"mG" }{TEXT 
-1 46 " using the binary method of exponentiation is " }{TEXT 263 16 "
Power(a,n) mod m" }{TEXT -1 33 ". We compare the time to compute " }
{XPPEDIT 18 0 "`mod`(a^n,m)" "6#-%$modG6$)%\"aG%\"nG%\"mG" }{TEXT -1 
70 " using the naive method as opposed to the binary method. First we \+
use " }{TEXT 297 16 "Power(2,n) mod m" }{TEXT -1 5 " for " }{XPPEDIT 
298 0 "n = 10^100;" "6#/%\"nG*$\"#5\"$+\"" }{TEXT -1 4 "and " }
{XPPEDIT 299 0 "m = 10^100-5;" "6#/%\"mG,&*$\"#5\"$+\"\"\"\"\"\"&!\"\"
" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "st:=tim
e():\nn:=10^100;\nm:=10^100 - 5;\n`2^n mod n` = Power(2,n) mod m;\nela
psed_time:=time() - st;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Now we
 compute using " }{TEXT 300 9 "2^n mod m" }{TEXT -1 10 " for just " }
{TEXT 301 4 "n = " }{XPPEDIT 302 0 "10^6;" "6#*$\"#5\"\"'" }{TEXT -1 
5 " and " }{XPPEDIT 303 0 "m = 10^6-5;" "6#/%\"mG,&*$\"#5\"\"'\"\"\"\"
\"&!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 
"restart:\nst:=time():\nn:=10^6;\nm:=10^6-5;\n`2^n mod m` = 2^n mod m;
\nelapsed_time:=time() - st;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "
You can see that the second method takes much longer even though the n
umbers involved are much smaller. In fact the second method would fail
 if we try  it for " }{XPPEDIT 18 0 "n = 10^100;" "6#/%\"nG*$\"#5\"$+
\"" }{TEXT -1 4 "and " }{XPPEDIT 18 0 "m = 10^100-5;" "6#/%\"mG,&*$\"#
5\"$+\"\"\"\"\"\"&!\"\"" }{TEXT -1 19 ", since the number " }{XPPEDIT 
18 0 "2^n;" "6#)\"\"#%\"nG" }{TEXT -1 64 " would be too large . It wou
ld have over 300,000 decimal places." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 46 "Converting \+
Text to Numbers and Numbers to Text" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 112 "We need to be able to convert text to numbers and numbers back
 to text to perform the RSA cryptographic scheme.\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "We can " }{TEXT 264 7 "convert
" }{TEXT -1 50 " text to a list of numbers by using the procedure " }
{TEXT 265 13 "convert/bytes" }{TEXT -1 270 " as in the following examp
les. Each  symbol (letters of the alphabet, punctuation mark, etc) cor
responds to a number from 1 to 255. Although the documentation does no
t say, this appears to be essentially the so-called ASCII (pronounced,
 \"ask-key\") code in decimal form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert(\"a\",bytes);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "convert([97],bytes);" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 204 "\nText should be
 enclosed in quotes \" \" . But note that \" within a text string will
 cause confusion unless you replace each \" by \\\", as in the followi
ng example.  First we show that the natural way fails.\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Text:=\"General Washington said, \"
Attack at dawn!\", before he went to sleep.\";" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 53 "Now we replace each \" inside the text string by  \\
\" :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Text:=\"General Was
hington said, \\\"Attack at dawn!\\\", before he went to sleep.\";" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 46 "We now convert this text to a list of numbers:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Numbers:=convert(Text,bytes);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Next we convert the numbers back t
o text:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert(Numbers,
bytes);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 123 "We can get a list of all symbols as follows.  We start w
ith a Maple command to obtain a list of the integers from 1 to 255." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "L:=[seq(i, i=1..255)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "N
ow we convert this list to symbols:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "L2:=convert(L,bytes);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "Now we convert back to numbers." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "convert(L2,bytes);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 
"" 0 "" {TEXT -1 90 "Converting to and from decimal, binary, octal and
 hexadecimal representations of integers." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 266 15 "convert(n, hex)" }{TEXT 
-1 95 " will convert the decimal number n to hexadecimal (that is, bas
e 16). Similarly we can replace " }{TEXT 304 4 "hex " }{TEXT -1 2 "by
" }{TEXT 305 7 " binary" }{TEXT -1 4 " or " }{TEXT 306 5 "octal" }
{TEXT -1 79 " to convert to binary (base 2) or octal (base 8). Let's c
onsider some examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "f
or n from  0 to 20 do \nn, convert(n,binary), convert(n,octal), conver
t(n,hex);\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 22 "convert(n,
decimal,hex)" }{TEXT -1 227 " will convert n from hexadecimal to decim
al. Similarly we can replace hex by binary or octal. Here are some exa
mples: Note that in the case of hexadecimal we need to place backquote
s about the number if it begins with a number." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "convert(A01234,decimal,hex);" }{TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "B
ut:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "convert(`123456789AB
CDEF`,decimal,hex);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "conv
ert(1234567,decimal,octal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "convert(1111111110000001111100000111,decimal,binary);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 22 "Homework Assignment #3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 276 21 "Format the worksheet:" }{TEXT -1 
42 " Open a new Maple worksheet. Click on the " }{TEXT 269 1 "T" }
{TEXT -1 156 " on the menu bar. This will allow you to insert text. Ne
xt go to the pull down menu on the left side where it says P Normal. P
ull this down till you get to " }{TEXT 270 7 "Title. " }{TEXT -1 9 "No
w type " }{TEXT 271 25 "Homework Assignment # 3. " }{TEXT -1 72 " Then
 hit return and type your name. Then click on the button that says " }
{TEXT 272 2 "[>" }{TEXT -1 32 " on the menu bar. Next click on " }
{TEXT 273 1 "T" }{TEXT -1 16 " again and type " }{TEXT 274 9 "problem \+
1" }{TEXT -1 16 ". Now click on  " }{TEXT 275 3 "[> " }{TEXT -1 19 " a
gain. Then solve " }{TEXT 277 9 "problem 1" }{TEXT -1 43 ". Similarly \+
label each problem in this way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 278 10 "problem 1." }{TEXT -1 48 "  Say that a n
umber whose binary representation " }{TEXT 287 9 "to base b" }{TEXT 
-1 23 " is [1,1,1,...,1] is a " }{TEXT 279 16 "base-b repunit. " }
{TEXT -1 46 "Thus a base-b repunit is a number of the form " }
{XPPEDIT 18 0 "(b^n-1)/(b-1);" "6#*&,&)%\"bG%\"nG\"\"\"F(!\"\"F(,&F&F(
F(F)F)" }{TEXT -1 118 ".  For each of the bases b = 2, 3, 4, 5, 10 fin
d the number of base-b repunits that are primes less than 1 google ( =
 " }{XPPEDIT 18 0 "10^100;" "6#*$\"#5\"$+\"" }{TEXT -1 2 ")." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 10 "problem 2." }
{TEXT -1 162 " The ISBN number for a book has 10 digits. The last is a
 check digit. It is obtained by the formula \n               (digit1*1
 + digit2*2 +....+ digit9*9) mod 11. " }}{PARA 0 "" 0 "" {TEXT -1 51 "
where digit1 is the left most digit of the number. " }{TEXT 288 3 "And
" }{TEXT -1 191 " if the check digit is 10 it is replaced by X. Look u
p the ISBN for two different books and use Maple to see that the check
 digits are correct. You might use the techniques in the section on " 
}{TEXT 281 31 "Addition Using for..do..end do." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "problem 3. " }{TEXT 
282 26 "Find all primes less than " }{XPPEDIT 18 0 "10^4;" "6#*$\"#5\"
\"%" }{TEXT -1 0 "" }{TEXT 289 11 " which are " }{TEXT 285 11 "palindr
omic" }{TEXT 286 50 ", i. e., they read the same forward as backwards.
 " }{TEXT -1 5 "Hint:" }{TEXT 291 106 " You may check whether a number
 n is palindromic by the following program: \n\nFirst we define the fu
nction " }{TEXT -1 3 "rev" }{TEXT 292 28 " which will reverse a list:
\n" }}{PARA 256 "> " 0 "" {MPLTEXT 1 0 35 "rev:=L->[seq(L[-i],i=1..nop
s(L))]:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Here's how to use it
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "rev([1,2,3,4]);" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 283 89 "Next you may use the following \+
idea to check whether or not an integer n is a palindrome:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 83 "n:=10109890101:\nL1:=convert(n,base,10);\ni
f L1 = rev(L1) then \n   print(n); \nend if;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 83 "n:=10109880101:\nL1:=convert(n,base,10);\nif L1 \+
= rev(L1) then \n   print(n); \nend if;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT 284 10 "problem 4." }{TEXT -1 10 " Find the " }{TEXT 290 
19 "hexadecimal, binary" }{TEXT -1 5 " and " }{TEXT 293 5 "octal" }
{TEXT -1 62 " representations of the palindromic primes found in probl
em 3." }}}}}{MARK "7" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }