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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 14 "Assignment 3  " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 10 "Problem 1." }
{TEXT -1 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 302 3 "(a)" }{TEXT -1 185 " Use a for-loop and a conditiona
l statement to split the positive integers not greater than 20 into a \+
set P of prime numbers and a set Q of non-prime numbers. Display the s
ets P and Q. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 303 3 "(b)" }{TEXT -1 49 " Use a for-loop and a conditional stat
ement with " }{TEXT 299 5 "break" }{TEXT -1 65 " to compute the smalle
st positive prime number greater than 1000." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 3 "(c)" }{TEXT -1 190 " Write a fo
r ... in loop that constructs from a set S a new set R consisting of e
ach element n in S divided by n+1. Apply it to the sets P and Q define
d in (a) above and display the results." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 11 "Problem 2. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 11 "Definiti
on." }{TEXT 279 25 " A positive integer n is " }{TEXT 269 7 "perfect" 
}{TEXT 280 49 " if the sum of its proper positive divisors is n." }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 104 "\nFor example, the prop
er positive divisors of 6 are 1, 2, and 3. Further 6 = 1 + 2 + 3. So 6
 is perfect." }}{PARA 0 "" 0 "" {TEXT 268 6 " \n(a) " }{TEXT -1 108 "W
rite a procedure which takes as input a positive integer n and returns
 the sum of all proper divisors of n." }{TEXT 262 1 " " }{TEXT -1 81 "
Show how your procedure works for the following values of n: 2, 3, 4, \+
6, 30, 900." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 4 "(b) " }{TEXT 263 34 "Using  the procedure you wrote in " }
{TEXT 281 3 "(a)" }{TEXT 282 1 " " }{TEXT -1 18 "write a procedure " }
{TEXT 283 9 "isperfect" }{TEXT -1 55 " which takes as input a positive
 integer n and returns " }{TEXT 260 4 "true" }{TEXT -1 21 " if n is pe
rfect and " }{TEXT 261 5 "false" }{TEXT -1 9 " if not. " }{TEXT 258 4 
"Note" }{TEXT -1 53 ": proper divisors of n can  not be greater than n
/2. " }{TEXT 271 46 "Use the procedure to find all perfect numbers " }
{XPPEDIT 18 0 "n <= 1000;" "6#1%\"nG\"%+5" }{TEXT 272 2 ". " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 17 "I
nteresting Fact:" }{TEXT 284 293 " It is an open question whether or n
ot there are any odd perfect numbers. So far no one has found one. In \+
number theory you learn the structure of the even perfect numbers. So \+
far only a finite number of even perfect numbers have been found. But \+
it is suspected that there are infinitely many." }{TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 264 11 "Problem 3 ." }{TEXT -1 3 "  \n" }}{PARA 0 "" 0 
"" {TEXT 285 10 "Definition" }{TEXT -1 6 ". The " }{TEXT 267 5 "order
" }{TEXT -1 1 " " }{TEXT 274 13 "of an integer" }{TEXT -1 3 " a " }
{TEXT 273 17 "modulo an integer" }{TEXT -1 59 " m is defined to be the
 least positive integer e such that " }{XPPEDIT 18 0 "`mod`(a^e,m) = 1
;" "6#/-%$modG6$)%\"aG%\"eG%\"mG\"\"\"" }{TEXT -1 29 ".  \n*(where e i
s an integer)*" }}{PARA 0 "" 0 "" {TEXT -1 28 "\nWrite a procedure, ca
ll it " }{TEXT 265 3 "Ord" }{TEXT -1 12 ", such that " }{TEXT 275 8 "O
rd(a,m)" }{TEXT -1 9 " returns " }{TEXT 290 4 "FAIL" }{TEXT -1 46 " if
 the order does not exist and the order of " }{TEXT 297 10 "a modulo m
" }{TEXT -1 18 " if  it  exists.\n\n" }{TEXT 286 91 "You may use the n
umber theoretic result that the order of a modulo m exists if and only
 if " }{TEXT 276 13 "igcd(a,m) = 1" }{TEXT 287 46 ". Also you may use \+
the fact that the order of " }{TEXT 291 10 "a modulo m" }{TEXT 292 23 
" is never greater than " }{TEXT 293 5 "m - 1" }{TEXT 294 24 ". [Note \+
that Maple uses " }{TEXT 277 4 "igcd" }{TEXT 288 49 " for the greatest
 common divisor of integers and " }{TEXT 278 3 "gcd" }{TEXT 289 46 " f
or greatest common divisor of polynomials.] " }{TEXT -1 2 "\n\n" }
{TEXT 300 4 "(a) " }{TEXT -1 23 "Check that you obtain: " }{TEXT 266 
65 "\n\n        Ord(0,11) = FAIL,    Ord(2,11) = 10,     Ord(4,11) = 5
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 3 "(b)
" }{TEXT -1 9 " Compute " }{TEXT 296 10 "Ord(a, 15)" }{TEXT -1 20 " fo
r a from 0 to 20." }}{PARA 0 "" 0 "" {TEXT 295 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 13 "Extra Credit." }{TEXT 
-1 3 "  \n" }}{PARA 0 "" 0 "" {TEXT -1 431 "Perhaps you have heard  th
e famous story about Ramanujan and Hardy's taxi number: Once when Rama
nujan was ill in London, Hardy visited Ramanujan in the hospital. When
 Hardy remarked that he had taken taxi number 1729, a singularly unexc
eptional number, Ramanujan immediately responded that this number was \+
actually quite remarkable: it is the smallest positive integer that ca
n be represented in two ways by the sum of two cubes. [" }{TEXT 306 
60 "from Eric Weisstein's Treasure Trove of Scientific Biography" }
{TEXT -1 1 "]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 428 "Verify (using Maple) Ramanujan's statement that 1729 is \+
the smallest positive integer that can be expressed in two different w
ays as a sum of two cubes. You should be able to use the ideas in the \+
section above on perfect squares and sums of squares. [Use of the anal
ogue of the procedure taking sums of squares, but here we are talking \+
of cubes instead of squares.  There are other ways to do the problem t
hat may occur to you.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 8 "- End - " }
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