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imes" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 43 "Some Famous Open Problem
s in Number Theory " }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Primes" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 259 2 "A \+
" }{TEXT 256 5 "prime" }{TEXT 260 103 " is a positive integer greater \+
than 1 that has no positive factors other than 1 and the number itself
. " }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 70 "          1 is not
 prime since we require primes to be greater that 1." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The 1st prime is 2." }}
{PARA 0 "" 0 "" {TEXT -1 21 " \nThe 2nd prime is 3." }}{PARA 0 "" 0 "
" {TEXT -1 34 "\n     4 = 2x2 so  4 is not prime. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The 3rd prime is 5." }}
{PARA 0 "" 0 "" {TEXT -1 35 "\n      6 = 2x3 so  6 is not prime. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The 4-th \+
prime is 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 28 "     8, 9, 10 are not prime." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 21 "The 5-th prime is 11." }}{PARA 0 "" 0 "
" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 18 "The maple command " }
{TEXT 257 11 "ithprime(n)" }{TEXT -1 13 " returns the " }{TEXT 258 1 "
n" }{TEXT -1 299 "-th prime. If you execute the following commands you
 will see this. To execute a command (statement below in red) place th
e cursor somewhere in the red and hit the return key. You will see the
 output in blue. The cursor will move to the next command in red and y
ou can hit return to execute it also." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "ithprime(1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "ithprime(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ithpr
ime(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ithprime(100);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The following command will prod
uce the sequence of the first 100 primes:" }{MPLTEXT 1 0 1 "\024" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(ithprim
e(i), i=1..100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Twin Primes" }}{EXCHG {PARA 0 "" 
0 "" {TEXT 264 38 "How close together can two primes be? " }{TEXT -1 
295 "Note that if n > 2 and n is even then n = 2k for some integer k >
 1, so an even integer n > 2 is never prime. Note also if n is odd the
n n + 1 is even.  So aside from the case n = 2, n+1 = 3 two consecutiv
e positive integers is never prime. \n\nSo if n > 2 and n is prime the
n n+1 is never prime. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 58 "However, it is possible for n+2 to be prime. For exa
mple, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 
"3 and 5 are primes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 20 "5 and 7 are primes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 22 "11 and 13 are primes. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "17 and 19 are primes. \+
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 65 "If  both n and n+2 are primes we say that
 \{n, n+2\}  is a pair of " }{TEXT 261 11 "twin primes" }{TEXT -1 41 "
. A famous conjecture is the following:\n\n" }{TEXT 262 22 "Twin Prime
s Conjecture" }{TEXT 265 2 ": " }{TEXT 263 42 "There infinite many pai
rs of twin primes. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 179 "Evidence seems to show that this conjecture is
 true, but so far no one has been able to prove it.  If you execute th
e following Maple command you will see more twin prime pairs. [" }
{TEXT 266 109 "Do not worry if you don't understand these commands! We
 will discuss them in more detail later in the course." }{TEXT -1 2 "]
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "for i from 1 to 50 d
o\n  if ithprime(i+1) = ithprime(i) + 2 then        print(\{ithprime(i
),ithprime(i+1)\});\n  fi;\nod;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 141 "Such computations have been carried to g
reat extent. For example it has been shown that there are  27,412,679 \+
twin prime pairs \{n, n+2\} with " }{XPPEDIT 18 0 "n+2 < 10^10;" "6#2,
&%\"nG\"\"\"\"\"#F&*$\"#5F)" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Gaps Between Pr
imes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 251 "How far apart can succesive primes be? As we shall see later i
n the course if n is any positive integer there is a stretch of n cons
ecutive numbers which contain no primes. For example, here we find two
 succesive primes with a gap of 112 between them." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "p:=ithprime
(31545);\nq:=ithprime(31546);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "
The following list of 112 consecutive integers contains no primes." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(p+1,i=0..111);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 21 "Goldbach's Conjecture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 258 "" 0 "" {TEXT 267 23 "Goldbach's Conjecture: " }
{TEXT 268 59 "Every even integer greater than 2 is a sum of two primes
.\n\n" }{TEXT 269 0 "" }{TEXT 270 0 "" }{TEXT 271 13 "For example, " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "4 = 2 + 2
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "6 = 3 \+
+ 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "8 =
 5 + 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 
"10 = 3 + 7 = 5 + 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 10 "12 = 5 + 7" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 19 "14 = 3 + 11 = 7 + 7" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "We can use Maple to find for even
 n > 2 primes p and q such that n = p + q provided n is not too big. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We ne
ed to use the command " }{TEXT 272 7 "isprime" }{TEXT -1 14 ". The com
mand " }{TEXT 273 10 "isprime(n)" }{TEXT -1 9 " returns " }{TEXT 274 
4 "true" }{TEXT -1 19 " if n is prime and " }{TEXT 275 5 "false" }
{TEXT -1 19 " if n is not prime." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "isprime(6);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "isprime(97);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 374 "Note that if n is some number and p is a prime such
 that q = n - p is a prime then n = p + q. We use this together with t
he Maple isprime command to find prime p and q such that n = p + q for
 all even integers n from 4 to 60. You will notice that not only is ev
ery even integer > 2 a sum of two primes, but as n gets larger it is a
 sum of two primes in many different ways.\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 184 "for n from 4 to 60 by 2 do\n printf(\"%2d \",n)
;\n for p from 1 to n/2 do\n  if isprime(p) and isprime(n-p) then \n  \+
 printf(\"= %2d + %2d \", p,n-p); \n  end if;\n end do;\nprintf(\"\\n
\");\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Primes of the Form " }{XPPEDIT 
18 0 "n^2+1;" "6#,&*$%\"nG\"\"#\"\"\"F'F'" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 73 "Other questions involve the existence of primes i
n special forms such as " }{XPPEDIT 18 0 "n^2+1;" "6#,&*$%\"nG\"\"#\"
\"\"F'F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "2^n-1;" "6#,&)\"\"#%\"nG\"
\"\"F'!\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "2^(2^n)+1;" "6#,&)\"
\"#)F%%\"nG\"\"\"F(F(" }{TEXT -1 47 ".  We discuss some open questions
 of this type." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT 276 45 "Are there infinitely many primes of the form " }
{XPPEDIT 18 0 "n^2+1" "6#,&*$%\"nG\"\"#\"\"\"F'F'" }{TEXT 277 2 "?\n" 
}}{PARA 258 "" 0 "" {TEXT 278 122 "It is believed that the answer is y
es. But a proof has not been found yet. We use Maple to find some prim
es of this form. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for n \+
from 1 to 100 do\nif isprime(n^2+1) then print(n,n^2,n^2+1); end if;\n
end do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 15 "Mersenne Primes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A \+
" }{TEXT 280 14 "Mersenne prime" }{TEXT -1 24 " is a prime of the form
 " }{XPPEDIT 18 0 "2^n-1" "6#,&)\"\"#%\"nG\"\"\"F'!\"\"" }{TEXT -1 89 
". We shall have more to say about these primes later in the course. T
he open question is:" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 279 42 
"Are there infinitely many Mersenne primes?" }}{PARA 0 "" 0 "" {TEXT 
-1 41 "\nHere we just find some Mersenne primes. " }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 86 "for n from 1 to 100 do\n  if isprime(2^n-1) \+
then \n   print(n, 2^n-1);\n  end if;\nend do;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Ferm
at Primes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 2 "A " }{TEXT 282 12 "Fermat prime" }{TEXT -1 24 " is a prime
 of the form " }{XPPEDIT 18 0 "2^(2*n)-1;" "6#,&)\"\"#*&F%\"\"\"%\"nGF
'F'F'!\"\"" }{TEXT -1 89 ". We shall have more to say about these prim
es later in the course. One open question is:" }}{PARA 0 "" 0 "" 
{TEXT -1 1 "\n" }{TEXT 281 40 "Are there infinitely many Fermat primes
?" }}{PARA 0 "" 0 "" {TEXT -1 39 "\nHere we just find some Fermat prim
es. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "for n from 0 to 6 d
o \n if isprime(2^(2^n)+1) then \n print(n,2^(2^n)+1);\n end if;\nend \+
do;" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 260 "" 0 "" {TEXT -1 160 "Actually these are the only Fermat
 primes that have been found. So it is also an open question whether o
r not there are more than just these five Fermat primes." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 19 "The 3x+1 Conjecture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 151 "For this problem we need the function f(
x) = x/2 if x is an even integer and f(x) = 3x + 1 if x is an odd inte
ger. This is defined in Maple as follows:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 51 "f:=x->if type(x,even) then x/2  else 3*x+1; end if:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "We now illustrate it:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(5);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 6 "f(16);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "f(8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(4);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 283 132 "The 3x+1 conjecture says that if n is any positive
 integer then the sequence f(n),  f(f(n)),  f(f(f(n))), .  . .  always
 contains 1." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 114 "For example, starting with n = 5 we obtain as we did a
bove \nf(5) = 16.\nf(f(5)) = f(16) = 8.\nf(f(f(5))) = f(8) = 4." }}
{PARA 0 "" 0 "" {TEXT -1 54 "f(f(f(f(5)))) = f(4) = 2.\nf(f(f(f(f(5)))
)) = f(2) = 1." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 114 "In Maple we can write (f@@k)(n) to obtain the resul
t of applying f   k times starting with n. For example we have:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "(f@@1)(5), (f@@2)(5), (f@@3)
(5),(f@@4)(5), (f@@5)(5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Now \+
we get the sequence of the first 20 iterates of f applied to 7 as foll
ows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "seq((f@@k)(7),k=1..
20);" }}}{PARA 0 "" 0 "" {TEXT -1 58 "Notice that the 16th iterate giv
es 1 when we start with 7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 86 "Experiments seems to confirm that the conjecture \+
is true, but it remains to be proved." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "The Prime R
epunit Conjecture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 89 "Numbers of the form 1, 11, 111, 1111, 11111, .... \+
all of whose digits are 1's are called " }{TEXT 284 10 "repunits. " }
{TEXT -1 4 "The " }{TEXT 289 4 "unit" }{TEXT -1 23 " part is for 1 and
 the " }{TEXT 290 3 "rep" }{TEXT -1 67 " is for repeated. In this case
 we have the following conjecture: \n\n" }{TEXT 285 25 "Prime Reunit C
onjecture: " }{TEXT 286 40 "There are infinitely many prime repunits" 
}{TEXT 287 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 56 "We create with the command below the repunit with n 1's.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Repunit:=n->(10^n-1)/9:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Repunit(1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Repunit(2);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "Repunit(3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "Repunit(30);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "
The only values of n for which " }{TEXT 288 10 "Repunit(n)" }{TEXT -1 
201 "  is known to be prime are when n is one of the following seven n
umbers: 2, 19, 23, 317, 1031, 49081, 86453. Nevertheless number theori
sts tend to think that there are probably infinitely many of them." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 15 "Perfect Numbers" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "To
 describe " }{TEXT 299 16 "perfect numbers " }{TEXT -1 37 "we need to \+
know what is meant by the " }{TEXT 300 8 "divisors" }{TEXT -1 56 " of \+
a positive integer n. These are simply the positive " }{TEXT 301 7 "fa
ctors" }{TEXT -1 45 " of n. This is provided by the Maple command " }
{TEXT 291 11 "divisors(n)" }{TEXT -1 98 ". To use this command we firs
t need to \"load\" the package \"numtheory\" using the following comma
nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numtheory):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "divisors(6);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "divisors(30);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 18 "The Maple command " }{TEXT 302 8 "sigma(n)" }
{TEXT -1 13 " returns the " }{TEXT 303 28 "sum of the positive divisor
s" }{TEXT -1 20 " of n.  For example:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "sigma(6);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "sigma(30);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "A \+
divisor of n is said to be a " }{TEXT 292 14 "proper divisor" }{TEXT 
-1 87 " if it does not equal n. So, for example, the proper divisors o
f 6 and 30 are given by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"divisors(6) minus \{6\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "divisors(30) minus \{30\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 
"Hence the " }{TEXT 305 26 "sum of the proper divisors" }{TEXT -1 20 "
  of n is given by  " }{TEXT 304 12 "sigma(n) - n" }{TEXT -1 4 ". \n\n
" }{TEXT 294 11 "Definition." }{TEXT -1 1 " " }{TEXT 295 36 " A positi
ve integer n is said to be " }{TEXT 293 7 "perfect" }{TEXT 296 54 " if
 n is equal to the sum of the proper divisors of n." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "Here we find a
ll perfect numbers up to 10000 and we print out the factorization  of \+
each number. Later we will discuss these in more detail.\n " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "for n from 1 to 10000 do\nif
 sigma(n) - n = n then print(n, ifactor(n)); fi;\nod;" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 297 11 "Conjecture:" }
{TEXT -1 44 " There are infinitely many perfect numbers?\n" }}{PARA 
261 "" 0 "" {TEXT 298 11 "Conjecture:" }{TEXT -1 34 " There are no odd
 perfect numbers?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 9 "Factoring" }}{PARA 3 "" 0 "" {TEXT -1 1 " \+
" }}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 306 36 "Is there a f
ast factoring algorithm?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 16 "Maple's command " }{TEXT 307 10 "ifactor(n)" }
{TEXT -1 218 " will factor small integers very rapidly. But for larger
 integers it is not so fast. And for really large numbers it might tak
e centuries to fact the number.\n\nHere are a few examples: First we f
actor a 17 digit number:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"ifactor(12345678987654321);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "N
ow we factor a 31 digit number. Be patient. It will take a little long
er." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ifactor(100000000000
3700030000000000111);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "We will discuss factoring in more \+
detail later." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 0 0 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }