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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 64 "Testing for prime numbers and f
actoring into a product of primes" }}{PARA 257 "" 0 "" {TEXT -1 11 "Dw
ight Lahr" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 
-1 65 "The function \"isprime,\" how it works (from the Help descripti
on):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 67 "
The function isprime is a probabilistic primality testing routine. " }
}{PARA 15 "" 0 "" {TEXT -1 490 "It returns false if n is shown to be c
omposite within one strong pseudo-primality test and one Lucas test an
d returns true otherwise. If isprime returns true, n is ``very probabl
y'' prime - see Knuth ``The art of computer programming'', Vol 2, 2nd \+
edition, Section 4.5.4, Algorithm P for a reference and H. Reisel, ``P
rime numbers and computer methods for factorization''. No counter exam
ple is known and it has been conjectured that such a counter example m
ust be hundreds of digits long. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 49 "Here is
 a test of a Fermat prime using \"isprime\":" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "n:=1;\nm:=2^n;\nf
ermat:=2^m+1;\n\n# True or False: 2^(2^n) + 1 is prime.\nisprime(ferma
t);\n\n# What are the factors of 2^(2^n) + 1 ?\nifactor(fermat);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 59 "Euclid's proo
f that there are an infinite number of primes:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "# Is the nu
mber formed in the proof prime?\n# Try it for some different primes.\n
\nW:=(2*3*5+1);\nisprime(W);\nifactor(W);" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 89 "Euclid's proof that there are \+
an infinite number of primes with the last prime p = 191:\n\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "# By trial and error, find \+
how many primes are greater than 5 and less than or equal to 191. The \+
answer is 40.\n# The upper limit 41 on the loop below is always set to
 be one more.\n\nprod:=2*3*5:\np:=5:\nfor i from 1 while i < 41 do\n  \+
 np:=nextprime(p);\n   prod := prod * np;\n   p:=np;\nend do:\nTheLast
Prime:=np;\nprod;\nW:=prod + 1;\nisprime(W);\n" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT -1 116 "We can also calculate the Greatest Common Divisor of two
 numbers; and the next prime and previous prime of a number:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "g
cd(89,144);\nisprime(191);\nprevprime(191);\nnextprime(89);\nithprime(
43);      #gives ith prime number\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "?isprime" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 
"?nextprime" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?prevprime" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?loop" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "20 0 0" 194 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }