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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 57 "Chapter 19: Primality Testing Using The Rabin-Miller Test" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "M
aple can quickly be used to test when a number is composite or very \"
probably\" prime using the Rabin-Miller test. First, we unassign all v
ariables:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }{TEXT 259 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT 261 10 "Example \+
1:" }}{PARA 0 "" 0 "" {TEXT -1 92 "Suppose we want to determine if 172
9 is composite. We first store this number as a variable:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "n := 1729;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"nG\"%H<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "A \+
Maple procedure is a prearranged collection of command that Maple exec
utes together. The next procedure, " }{TEXT 262 4 "div2" }{TEXT -1 
136 ", is designed to determine to number of factors of 2 that a numbe
r has. By hitting return, this procedure will be read into the session
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "div2 := proc(num)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "   local k, nd2;" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "   k := 0; nd2 := num; " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "   while nd2 mod 2 = 0 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "      k := k+1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"      nd2 := nd2/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "   lprint(cat(\"Power 2 factorizat
ion of \", convert(num, string), \"= \", convert(ifactor(2^k), string)
, \"*\", convert(nd2, string)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "
   k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 3 "If " }{TEXT 263 1 "n" }{TEXT -1 101 " is the prime nu
mber candidate, the Rabin-Miller test requires getting the number of f
actors of 2 in " }{TEXT 268 1 "n" }{TEXT -1 20 " - 1, that is, write" 
}}{PARA 0 "" 0 "" {TEXT 265 1 "n" }{TEXT -1 7 " - 1 = " }{XPPEDIT 18 
0 "2^k;" "6#)\"\"#%\"kG" }{TEXT 266 1 "q" }{TEXT -1 8 ", where " }
{TEXT 267 1 "k" }{TEXT -1 32 " is an non-negative integer and " }
{TEXT 264 1 "q" }{TEXT -1 43 " is an odd number. This command will pri
nt " }{TEXT 269 2 "n " }{TEXT -1 64 "- 1's factorization and will outp
ut the number of factors of 2, " }{TEXT 270 1 "k" }{TEXT -1 23 ", that
 occur. We store " }{TEXT 271 1 "k" }{TEXT -1 30 " by entering the nex
t command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "k := div2(n-1
);" }}{PARA 6 "" 1 "" {TEXT -1 43 "\"Power 2 factorization of 1728= ``
(2)^6*27\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "The next command takes the previous resul
t and stores the odd integer " }{TEXT 272 1 "q" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "q := (n -1)/2^k;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"qG\"#F" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 25 "Given an integer witness " }{TEXT 277 1 "a" }{TEXT -1 66 ", the
 Rabin Miller Test requires computing the following sequence:" }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "a^q;" "6#)%\"aG%\"qG" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "a^(2*q);" "6#)%\"aG*&\"\"#\"\"\"%\"qGF'" }{TEXT -1 
2 ", " }{XPPEDIT 18 0 "a^(4*q);" "6#)%\"aG*&\"\"%\"\"\"%\"qGF'" }
{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "a^(2^(k-1)*q);" "6#)%\"aG*&)\"\"#
,&%\"kG\"\"\"F*!\"\"F*%\"qGF*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 66 "The next Maple gives the symbolic representation of this \+
sequence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "[seq( a^(ifact
or(2^(i-1))*q), i = 1..k)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(*$)%
\"aG\"#F\"\"\")F&,$*&F'F(-%!G6#\"\"#F(F()F&,$*&F'F()F,F/F(F()F&,$*&F'F
()F,\"\"$F(F()F&,$*&F'F()F,\"\"%F(F()F&,$*&F'F()F,\"\"&F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "For a number to be composite, the \+
Rabin-Miller test says that a witness " }{TEXT 279 1 "a" }{TEXT -1 32 
" must satisfy the the following:" }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " 
}{XPPEDIT 18 0 "a^q;" "6#)%\"aG%\"qG" }{TEXT -1 27 " is not congruent \+
to 1 mod " }{TEXT 273 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 3 "2. " }{XPPEDIT 18 0 "a^(2^i*q);" "6#)%\"aG*&)\"\"#%\"iG\"\"\"%\"
qGF)" }{TEXT -1 28 " is not congruent to -1 mod " }{TEXT 275 1 "n" }
{TEXT -1 9 " for all " }{TEXT 276 1 "i" }{TEXT -1 14 " = 0, 1, ..., " 
}{TEXT 274 1 "k" }{TEXT -1 3 "-1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 29 "Suppose, we want to check if " }{TEXT 
278 1 "a" }{TEXT -1 56 " = 2 is a Rabin-Miller witness. This command s
hows that " }{XPPEDIT 18 0 "a^q;" "6#)%\"aG%\"qG" }{TEXT -1 5 " mod " 
}{TEXT 280 2 "n " }{TEXT -1 29 "satisfies the first criterea." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(a = 2, a)&^ q mod n;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$X'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "Since " }{XPPEDIT 18 0 "a^q;" "6#)%\"aG%\"qG" }{TEXT -1 5 
" mod " }{TEXT 281 2 "n " }{TEXT -1 26 "is not congruent to 1 mod " }
{TEXT 282 3 "n, " }{TEXT -1 37 "we proceed to criterea 2. We use the \+
" }{TEXT 283 5 "mods " }{TEXT -1 42 "command to do the modular arithme
tic. The " }{TEXT 284 4 "mods" }{TEXT -1 108 " does symmetric represen
tation modular arithmetic, that is, integer remainders are computed us
ing a modulus " }{TEXT 322 1 "m" }{TEXT -1 28 " are found in the inter
val [" }{TEXT 285 1 "-" }{TEXT -1 1 "(" }{TEXT 289 2 "m-" }{TEXT -1 4 
"1)/2" }{TEXT 287 3 ", m" }{TEXT -1 19 "/2] instead of [0.." }{TEXT 
286 1 "m" }{TEXT -1 110 "-1] as is done in standard modular arithmetic
. This will allow the -1 remainder representation to be used for " }
{TEXT 288 1 "m" }{TEXT -1 87 "-1. To see that none of the powers of th
e witness candidate is not congruent to -1 mod " }{TEXT 290 3 "n, " }
{TEXT -1 5 "enter" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "rmill \+
:= [ seq( mods(subs(a = 2, a)&^(2^(i-1)*q), n), i = 1..k) ] ;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rmillG7(\"$X'!$k'\"\"\"F(F(F(" }}}
{EXCHG {PARA 257 "" 1 "" {TEXT -1 6 "Thus, " }{TEXT 291 1 "a" }{TEXT 
-1 28 " = 2 will be a witness that " }{TEXT 294 1 "n" }{TEXT -1 51 " =
 1729 is composite. We can verify this using the " }{TEXT 295 7 "ifact
or" }{TEXT -1 9 " command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "ifactor(1729);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%!G6#\"\"(\"
\"\"-F%6#\"#8F(-F%6#\"#>F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 292 0 "" }{TEXT 293 10 "Example 2
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Supp
ose now that we want to test whether " }{TEXT 296 1 "n" }{TEXT -1 50 "
 = 172947529 is composite. We store this value of " }{TEXT 297 1 "n" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "n := 17294
7529;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"*Hv%H<" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 24 "The next command stores " }{TEXT 298 2 "k
 " }{TEXT -1 4 "and " }{TEXT 299 1 "q" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "k := div2(n-1); q := (n-1)/2^k;" }}
{PARA 6 "" 1 "" {TEXT -1 54 "\"Power 2 factorization of 172947528= ``(
2)^3*21618441\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG\")T%=;#" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 58 "This is a symbolic representation of the witness seq
uence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "[seq( a^(ifactor(
2^(i-1))*q), i = 1..k)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$)%\"aG
\")T%=;#\"\"\")F&,$*&F'F(-%!G6#\"\"#F(F()F&,$*&F'F()F,F/F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Not all choices for" }{TEXT 300 2 
" a" }{TEXT -1 100 " represent a Rabin-Miller witness. The next two co
mmands demonstrate this for the witness candidate " }{TEXT 301 1 "a" }
{TEXT -1 5 " = 3." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(a
 = 3, a)&^ q mod n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"*Gv%H<" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rmill := [seq( mods(subs(a =
 3, a)&^(2^(i-1)*q), n), i = 1..k) ] ;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&rmillG7%!\"\"\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 
"Since there is a congruence to -1 in the witness sequence, " }{TEXT 
302 1 "a" }{TEXT -1 33 " = 3 is not a Rabin Witness that " }{TEXT 323 
1 "n" }{TEXT -1 63 " is composite. However, the next two commands demo
nstrate that " }{TEXT 303 1 "a" }{TEXT -1 8 " = 2 is." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(a = 2, a)&^ q mod n;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\")1Q1S" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "rmill := [seq( mods(subs(a = 2, a)&^(2^(i-1)*q), n), \+
i = 1..k) ] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rmillG7%\")1Q1S\"(
lqD#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Thus, " }{TEXT 304 
1 "n" }{TEXT -1 52 " = 172947529 is composite as the next command show
s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(n);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#*(-%!G6#\"$2$\"\"\"-F%6#\"$8'F(-F%6#\"$>*F(
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 305 0 "" }{TEXT 306 10 "Example 3:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "We next consider \+
the integer " }{TEXT 316 1 "n" }{TEXT -1 56 " = 5967986236329763296324
24696281. We store this number." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "n := 596798623632976329632424696281;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"nG\"?\"G'pCCjHj(HjB')z'f" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 35 "We first find the number of 2's in " }{TEXT 307 1 "n" }
{TEXT -1 29 "-1's factorization and store " }{TEXT 308 1 "q" }{TEXT 
-1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "k := div2(n-1); \+
q := (n-1)/2^k;" }}{PARA 6 "" 1 "" {TEXT -1 96 "\"Power 2 factorizatio
n of 596798623632976329632424696280= ``(2)^3*7459982795412204120405308
7035\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"$" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"qG\">Nq3`S?T?7az#)*fu" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 60 "This is the symbolic representation of the witness seq
uence." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "[seq( a^(ifactor(
2^(i-1))*q), i = 1..k)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$)%\"aG
\">Nq3`S?T?7az#)*fu\"\"\")F&,$*&F'F(-%!G6#\"\"#F(F()F&,$*&F'F()F,F/F(F
(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The next set of commands sho
ws that " }{TEXT 309 1 "a" }{TEXT -1 68 " = 2, 3, 4, and 5 fail to be \+
Rabin-Miller witnesses for this number." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 310 1 "a" }{TEXT -1 5 " = 2:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(a = 2, a)&^ q mod n; " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"?()3***Gf/z*p,QXFaP" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rmill := [seq( mods(subs(a = 2, a)&
^(2^(i-1)*q), n), i = 1..k) ] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
rmillG7%!?%R0Z8t\"RlFJ)p6P@#!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 1 "a" }{TEXT -1 5 " = 3:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(a = 3, a)&^ q mod n; \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\">-4n>)Huq!fStF\"yY" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rmill := [seq( mods(subs(a = 3, a)&
^(2^(i-1)*q), n), i = 1..k) ] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
rmillG7%\">-4n>)Huq!fStF\"yY\"?%R0Z8t\"RlFJ)p6P@#!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 1 "a" }
{TEXT -1 5 " = 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(a
 = 4, a)&^ q mod n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"?!G'pCCjHj(Hj
B')z'f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rmill := [seq( mo
ds(subs(a = 4, a)&^(2^(i-1)*q), n), i = 1..k) ] ;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&rmillG7%!\"\"\"\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 1 "a" }{TEXT -1 5 " = 5:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(a = 5, a)&^ q mod n;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"?%R0Z8t\"RlFJ)p6P@#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "rmill := [seq( mods(subs(a = 5, a)&
^(2^(i-1)*q), n), i = 1..k) ] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&
rmillG7%\"?%R0Z8t\"RlFJ)p6P@#!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We might start to assert \+
that this number might be prime. In fact, for " }{TEXT 314 1 "t" }
{TEXT -1 90 " failed Rabin-Witness candidates, the probability that n \+
is prime is given by the formula:" }}{PARA 256 "" 0 "" {TEXT -1 2 "1-
" }{XPPEDIT 18 0 ".75^t;" "6#)-%&FloatG6$\"#v!\"#%\"tG" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 23 "Thus, for the above 4 (" }{TEXT 315 
1 "a" }{TEXT -1 87 " = 2..5) failed witness candidates, we have shown \+
that the probability that the number " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 317 1 "n" }{TEXT -1 45 " = 596798623632976329632424696281 is \+
prime is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "1 - .75^4;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")v$f$o!\")" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 249 "Thus, there is a about a 68.4 % chance this number is
 prime. We can increase our chances by testing more witnesses. The nex
t command shows approximately how many failed witness candidates are n
eeded to show n is a prime for a probability of 99.999%." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(1 - .75^t = .99999, t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!Rh>+%!\")" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 82 "Thus, we need 41 non-witness candidates to assure th
is probability. Using a Maple " }{TEXT 318 3 "for" }{TEXT -1 21 " loop
 along with the " }{TEXT 319 3 "seq" }{TEXT -1 47 " command, we can qu
ickly test these witnesses. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "for j from 2 to 42 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "   l
print(cat(\"a = \", convert(j, string), \"=> a^q = \", convert(subs(a \+
= j, a)&^ q mod n, string)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "l
print(cat(\"a^(2^i*q) = \", convert([seq( mods(subs(a = j, a)&^(2^(i-1
)*q), n), i = 1..k) ], string))); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "print(\" \");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 6 
"" 1 "" {TEXT -1 46 "\"a = 2=> a^q = 375427453801699790459289990887\"
" }}{PARA 6 "" 1 "" {TEXT -1 54 "\"a^(2^i*q) = [-221371169831276539173
134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}
{PARA 6 "" 1 "" {TEXT -1 45 "\"a = 3=> a^q = 4678127734059070742981967
0902\"" }}{PARA 6 "" 1 "" {TEXT -1 81 "\"a^(2^i*q) = [4678127734059070
7429819670902, 221371169831276539173134705394, -1]\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 46 "\"a = 4=> a^
q = 596798623632976329632424696280\"" }}{PARA 6 "" 1 "" {TEXT -1 24 "
\"a^(2^i*q) = [-1, 1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"
" }}{PARA 6 "" 1 "" {TEXT -1 46 "\"a = 5=> a^q = 221371169831276539173
134705394\"" }}{PARA 6 "" 1 "" {TEXT -1 53 "\"a^(2^i*q) = [22137116983
1276539173134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6
\"" }}{PARA 6 "" 1 "" {TEXT -1 46 "\"a = 6=> a^q = 3455276089085096413
42190300287\"" }}{PARA 6 "" 1 "" {TEXT -1 84 "\"a^(2^i*q) = [-25127101
4724466688290234395994, -221371169831276539173134705394, -1]\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 
46 "\"a = 7=> a^q = 345527608908509641342190300287\"" }}{PARA 6 "" 1 "
" {TEXT -1 84 "\"a^(2^i*q) = [-251271014724466688290234395994, -221371
169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"
~6\"" }}{PARA 6 "" 1 "" {TEXT -1 46 "\"a = 8=> a^q = 22137116983127653
9173134705394\"" }}{PARA 6 "" 1 "" {TEXT -1 53 "\"a^(2^i*q) = [2213711
69831276539173134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q
\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 46 "\"a = 9=> a^q = 221371169831276
539173134705394\"" }}{PARA 6 "" 1 "" {TEXT -1 53 "\"a^(2^i*q) = [22137
1169831276539173134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 18 "\"a = 10=> a^q = 1\"" }}
{PARA 6 "" 1 "" {TEXT -1 23 "\"a^(2^i*q) = [1, 1, 1]\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 46 "\"a = 11=>
 a^q = 46781277340590707429819670902\"" }}{PARA 6 "" 1 "" {TEXT -1 81 
"\"a^(2^i*q) = [46781277340590707429819670902, 22137116983127653917313
4705394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "
" 1 "" {TEXT -1 47 "\"a = 12=> a^q = 550017346292385622202605025379\"
" }}{PARA 6 "" 1 "" {TEXT -1 82 "\"a^(2^i*q) = [-467812773405907074298
19670902, 221371169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 13=> a^q \+
= 550017346292385622202605025379\"" }}{PARA 6 "" 1 "" {TEXT -1 82 "\"a
^(2^i*q) = [-46781277340590707429819670902, 22137116983127653917313470
5394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 
1 "" {TEXT -1 47 "\"a = 14=> a^q = 550017346292385622202605025379\"" }
}{PARA 6 "" 1 "" {TEXT -1 82 "\"a^(2^i*q) = [-467812773405907074298196
70902, 221371169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 15=> a^q \+
= 251271014724466688290234395994\"" }}{PARA 6 "" 1 "" {TEXT -1 83 "\"a
^(2^i*q) = [251271014724466688290234395994, -2213711698312765391731347
05394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 
1 "" {TEXT -1 18 "\"a = 16=> a^q = 1\"" }}{PARA 6 "" 1 "" {TEXT -1 23 
"\"a^(2^i*q) = [1, 1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"
" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 17=> a^q = 55001734629238562220
2605025379\"" }}{PARA 6 "" 1 "" {TEXT -1 82 "\"a^(2^i*q) = [-467812773
40590707429819670902, 221371169831276539173134705394, -1]\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 18 "\"a \+
= 18=> a^q = 1\"" }}{PARA 6 "" 1 "" {TEXT -1 23 "\"a^(2^i*q) = [1, 1, \+
1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" 
{TEXT -1 47 "\"a = 19=> a^q = 550017346292385622202605025379\"" }}
{PARA 6 "" 1 "" {TEXT -1 82 "\"a^(2^i*q) = [-4678127734059070742981967
0902, 221371169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 20=> a^q \+
= 375427453801699790459289990887\"" }}{PARA 6 "" 1 "" {TEXT -1 54 "\"a
^(2^i*q) = [-221371169831276539173134705394, -1, 1]\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 18 "\"a = 21=>
 a^q = 1\"" }}{PARA 6 "" 1 "" {TEXT -1 23 "\"a^(2^i*q) = [1, 1, 1]\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 
47 "\"a = 22=> a^q = 345527608908509641342190300287\"" }}{PARA 6 "" 1 
"" {TEXT -1 84 "\"a^(2^i*q) = [-251271014724466688290234395994, -22137
1169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q
\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 23=> a^q = 55001734629238
5622202605025379\"" }}{PARA 6 "" 1 "" {TEXT -1 82 "\"a^(2^i*q) = [-467
81277340590707429819670902, 221371169831276539173134705394, -1]\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 
47 "\"a = 24=> a^q = 251271014724466688290234395994\"" }}{PARA 6 "" 1 
"" {TEXT -1 83 "\"a^(2^i*q) = [251271014724466688290234395994, -221371
169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"
~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 25=> a^q = 5967986236329763
29632424696280\"" }}{PARA 6 "" 1 "" {TEXT -1 24 "\"a^(2^i*q) = [-1, 1,
 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" 
{TEXT -1 47 "\"a = 26=> a^q = 251271014724466688290234395994\"" }}
{PARA 6 "" 1 "" {TEXT -1 83 "\"a^(2^i*q) = [25127101472446668829023439
5994, -221371169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 27=> a^q \+
= 251271014724466688290234395994\"" }}{PARA 6 "" 1 "" {TEXT -1 83 "\"a
^(2^i*q) = [251271014724466688290234395994, -2213711698312765391731347
05394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 
1 "" {TEXT -1 47 "\"a = 28=> a^q = 251271014724466688290234395994\"" }
}{PARA 6 "" 1 "" {TEXT -1 83 "\"a^(2^i*q) = [2512710147244666882902343
95994, -221371169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 29=> a^q \+
= 550017346292385622202605025379\"" }}{PARA 6 "" 1 "" {TEXT -1 82 "\"a
^(2^i*q) = [-46781277340590707429819670902, 22137116983127653917313470
5394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 
1 "" {TEXT -1 46 "\"a = 30=> a^q = 46781277340590707429819670902\"" }}
{PARA 6 "" 1 "" {TEXT -1 81 "\"a^(2^i*q) = [46781277340590707429819670
902, 221371169831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 31=> a^q \+
= 596798623632976329632424696280\"" }}{PARA 6 "" 1 "" {TEXT -1 24 "\"a
^(2^i*q) = [-1, 1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}
{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 32=> a^q = 375427453801699790459289
990887\"" }}{PARA 6 "" 1 "" {TEXT -1 54 "\"a^(2^i*q) = [-2213711698312
76539173134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"
" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 33=> a^q = 22137116983127653917
3134705394\"" }}{PARA 6 "" 1 "" {TEXT -1 53 "\"a^(2^i*q) = [2213711698
31276539173134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~
6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 34=> a^q = 25127101472446668
8290234395994\"" }}{PARA 6 "" 1 "" {TEXT -1 83 "\"a^(2^i*q) = [2512710
14724466688290234395994, -221371169831276539173134705394, -1]\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 
46 "\"a = 35=> a^q = 46781277340590707429819670902\"" }}{PARA 6 "" 1 "
" {TEXT -1 81 "\"a^(2^i*q) = [46781277340590707429819670902, 221371169
831276539173134705394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6
\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 36=> a^q = 375427453801699790
459289990887\"" }}{PARA 6 "" 1 "" {TEXT -1 54 "\"a^(2^i*q) = [-2213711
69831276539173134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q
\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 37=> a^q = 37542745380169
9790459289990887\"" }}{PARA 6 "" 1 "" {TEXT -1 54 "\"a^(2^i*q) = [-221
371169831276539173134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 47 "\"a = 38=> a^q = 2512710147
24466688290234395994\"" }}{PARA 6 "" 1 "" {TEXT -1 83 "\"a^(2^i*q) = [
251271014724466688290234395994, -221371169831276539173134705394, -1]\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT 
-1 47 "\"a = 39=> a^q = 375427453801699790459289990887\"" }}{PARA 6 "
" 1 "" {TEXT -1 54 "\"a^(2^i*q) = [-221371169831276539173134705394, -1
, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" 
{TEXT -1 47 "\"a = 40=> a^q = 596798623632976329632424696280\"" }}
{PARA 6 "" 1 "" {TEXT -1 24 "\"a^(2^i*q) = [-1, 1, 1]\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 6 "" 1 "" {TEXT -1 46 "\"a = 41
=> a^q = 46781277340590707429819670902\"" }}{PARA 6 "" 1 "" {TEXT -1 
81 "\"a^(2^i*q) = [46781277340590707429819670902, 22137116983127653917
3134705394, -1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}{PARA 
6 "" 1 "" {TEXT -1 47 "\"a = 42=> a^q = 375427453801699790459289990887
\"" }}{PARA 6 "" 1 "" {TEXT -1 54 "\"a^(2^i*q) = [-2213711698312765391
73134705394, -1, 1]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"~6\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Thus, " }{TEXT 320 1 "n" }{TEXT -1 
91 " = 596798623632976329632424696281 is almost certainly prime. This \+
is agrees with the Maple " }{TEXT 321 7 "isprime" }{TEXT -1 9 " comman
d." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "isprime(5967986236329
76329632424696281);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 2" 10 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }