{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 0 12 0 0 128 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Tex t Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# floyd_cy.ms\n" } {TEXT -1 723 "## Is 2^p-1 prime for infinitely many p? ##\n# I s 2^(2^n)+1 prime for some n > 4? #\n# \+ #\n# John B. Cosgrave, #\n # Department of Mathematics, #\n# St. Patric k's College, #\n# Drumcondra, \+ #\n# Dublin 9, #\n# IRE LAND. #\n# \+ #\n# Home e-mail: johnbcos@iol.ie #\n# Coll ege e-mail: John.Cosgrave@spd.ie #\n# \+ #\n# 2^sqrt(2) is transcendental. Pi(x)~x/ln(x). # \+ \n## .12345678910111213 ... is normal. Is Pi? ##\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "This worksheet is about the Pollard-rho f actoring method, using the Floyd cycle algorithm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "'n' is known to be compos ite because of ... ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 23 "What one wants is this:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 98 "a[1], a[2], a[3], a[4], ... defined b y Pollard suggestion, and one then wants to form the numbers:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " \+ gcd(n, a[2] - a[1])," }}{PARA 0 "" 0 "" {TEXT -1 43 " \+ gcd(n, a[4] - a[2])," }}{PARA 0 "" 0 "" {TEXT -1 53 " gcd(n, a[6] - a[3]), etc." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "One wants to get a n output NOT equal to 1 ... " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "This was the example I began with:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 10 "a[1] := 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a[2] := a[1]^2 + 1 mod 1037:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for k from 2 while igcd(1037, a[2*k-2] - a[k-1]) = 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a[k] := a[k-1]^2 + 1 mod 1037;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a[2*k] := (a[2*k-2]^2 + 1 mod 1037)^2 + 1 mod 1037;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seq(igcd(a[2*K] - a[K], 10 37), K=1..k-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a[1] := 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a[2] := a[1]^2 + 1 mod 1037:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for k from 2 while igcd(1037, a[2*k -2] - a[k-1]) = 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a[k] := a[k -1]^2 + 1 mod 1037;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a[2*k] := (a [2*k-2]^2 + 1 mod 1037)^2 + 1 mod 1037;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "igcd(a[2*k-2] - a[k-1], 1037);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "standard_Pollard_Floyd := pr oc(n) # using Pollard's (x^2 + 1) as default function" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "local a, k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a[1] := 2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a[2] := a[1]^2 + \+ 1 mod n:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "for k from 2 while igcd (n, a[2*k-2] - a[k-1]) = 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "a[ k] := a[k-1]^2 + 1 mod n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a[2*k] := (a[2*k-2]^2 + 1 mod n)^2 + 1 mod n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "igcd(a[2*k-2] - a[k-1 ], n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 2 "" 1 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "standard_ Pollard_Floyd(91);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "stand ard_Pollard_Floyd(2^32 - 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "standard_Pollard_Floyd(2^32 + 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "gen_P_F := proc(n, f, seed) # general Pollard-Floyd" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "local a, k;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "a[1] := seed:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a[2] := f(a[1] ) mod n:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "for k from 2 while igcd (n, a[2*k-2] - a[k-1]) = 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a[ k] := f(a[k-1]) mod n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a[2*k] := f(f(a[2*k-2])) mod n;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "if igcd(a[2*k-2] - a[k-1], n) < n t hen" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "lprint(igcd(a[2*k-2] - a[k-1 ], n), `is a proper factor of `, n) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "else lprint(`Try some other seed or function.`) " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 2 "" 1 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gen_P_F (1037, x -> x^2 - 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gen_P_F(1037, x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gen_P_F(1037, x -> x^2, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gen_P_F(1037, x -> x^2 + 2, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gen_P_F(1037, x -> x^2 + 3, 2);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gen_P_F(1037, x -> x^3 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gen_P_F(1037, x -> \+ x^3 + 2, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "gen_P_F(2^32 + 1, x -> x^2 + 1 mod \+ (2^32 + 1), 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "gen_P_F( 2^64 + 1, x -> x^2 + 1 mod (2^64 + 1), 2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "gen_P_F(2^128 + 1, x -> x&^1024 + 1 mod (2^128 + 1) , 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# gen_P_F(2^128 + \+ 1, x -> x&^1024 + 1 mod (2^128 + 1), 3);" }}{PARA 2 "" 1 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Abandoned after 10 hours ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Now \+ to have a look at numbers that are like Fermat numbers [but which beha ve quite differently!!]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "gen_P_F (2^16 + 2^8 + 1, x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gen_P_F(2^32 + 2^16 + 1, x -> x^2 + 1, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "gen_P_F(2^64 + 2^32 + 1, x -> x^2 + 1, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "gen_P_F((2^64 + 2^32 + 1)/3, x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "gen_P _F((2^64 + 2^32 + 1)/(3*7), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "gen_P_F((2^64 + 2^32 + 1)/(3*7*97), x -> x^2 + 1 , 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "gen_P_F((2^64 + 2^ 32 + 1)/(3*7*97*13), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isprime(673);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "gen_P_F((2^64 + 2^32 + 1)/(3*7*97*13*673), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "isprime(241);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "gen_P_F((2^64 + 2^32 + 1)/(3*7*97*1 3*673*241), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "gen_P_F((2^64 + 2^32 + 1)/(3*7*97*13*673*241*193), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "(2^64 + 2^32 + 1)/( 3*7*97*13*673*241*193);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " isprime(22253377);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "2^64 \+ + 2^32 + 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Thus the prime factorisation of (2^64 + 2 ^32 + 1) is 3*7*13*97*241*193*673*22253377." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ifactor(2^64 + 2^32 + 1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Let us now \+ attempt to factor the number:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "2^ 128 + 2^64 + 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "gen_P_F( 2^128 + 2^64 + 1, x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "gen_P_F((2^128 + 2^64 + 1)/3, x -> x^2 + 1, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "gen_P_F((2^128 + 2^64 + 1)/( 3*7), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " gen_P_F((2^128 + 2^64 + 1)/(3*7*97), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "gen_P_F((2^128 + 2^64 + 1)/(3*7*97* 13), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "g en_P_F((2^128 + 2^64 + 1)/(3*7*97*13*673), x -> x^2 + 1, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "gen_P_F((2^128 + 2^64 + 1)/( 3*7*97*13*673*241), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "gen_P_F((2^128 + 2^64 + 1)/(3*7*97*13*673*241*193), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "isprime ((2^128 + 2^64 + 1)/(3*7*97*13*673*241*193*22253377));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "(2^128 + 2^64 + 1)/(3*7*97*13*673*2 41*193*22253377);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "And so note that while the factor isation of (2^128 + 1) caused a real problem, the factorisation of " } }{PARA 0 "" 0 "" {TEXT -1 128 "(2^128 + 2^64 + 1) was straightforward. [There are - of course - other things to be noted (and explained): 3 , 7, 97, 13, ... ] " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 40 "Let us now attempt to factor the number:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "2^256 + 2^128 + 1;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "gen_P_F(2^256 + 2^128 + 1, x -> x^2 + 1, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "gen_P_F((2^256 + 2^128 + 1)/ 3, x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "gen _P_F((2^256 + 2^128 + 1)/(3*7), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 55 "gen_P_F((2^256 + 2^128 + 1)/(3*7*97), x -> x ^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "gen_P_F((2^2 56 + 2^128 + 1)/(3*7*97*13), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 62 "gen_P_F((2^256 + 2^128 + 1)/(3*7*97*13*673), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "gen_P_F ((2^256 + 2^128 + 1)/(3*7*97*13*673*241), x -> x^2 + 1, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "gen_P_F((2^256 + 2^128 + 1)/ (3*7*97*13*673*241*769), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 90 "# gen_P_F((2^256 + 2^128 + 1)/(3*7*97*13*673*241*76 9*193), x -> x^2 + 1, 2); STOPPED ... " }}{PARA 2 "" 1 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "ifactor((2^256 + 2^1 28 + 1)/(3*7*97*13*673*241*769*193), easy);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "factor(x^32 + x^16 + 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 " (2^256 + 2^128 + 1) factors (algebraically) as:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " (2^2 - 2 + 1)*(2^2 + 2 + 1)*(2^4 - 2^2 + 1)*(2^8 - 2^4 + 1)*(2^16 - 2^8 + 1)" }}{PARA 0 "" 0 "" {TEXT -1 56 "*(2^32 - 2^16 + 1)*(2^64 - 2^32 + 1)*(2^128 - 2^64 + 1)." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "(2^2 - 2 + 1); (2^2 + 2 + 1); (2^4 \+ - 2^2 + 1); (2^8 - 2^4 + 1); (2^16 - 2^8 + 1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "(2^256 + 2^128 + 1) mod 65281;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ifactor(65281);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "The '769' and '193' divide other terms:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "(2^128 - 2^64 + 1) mod 769; " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "(2^32 - 2^16 + 1) mod 193; " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "gen_P_F(2^64 + 2^16 + 1, x -> x^2 + 1, 2);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "gen_P_F((2^64 + 2^16 + 1)/3 , x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ispr ime(5669);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "gen_P_F((2^64 + 2^16 + 1)/(3*5669), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "isprime(17207);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "gen_P_F ((2^128 + 2^32 + 1), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "gen_P_F((2^128 + 2^32 + 1)/3, x -> x^2 + 1, 2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "gen_P_F((2^128 + 2^32 + 1)/( 3*3), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 " gen_P_F((2^128 + 2^32 + 1)/(3*3*3), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "gen_P_F((2^128 + 2^32 + 1)/(3*3*3*3 ), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "NICE!!!" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " isprime(80271439);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "gen_P_F((2^128 + 2^32 + 1)/(3*3*3*3 *80271439), x -> x^2 + 1, 2);" }}{PARA 2 "" 1 "" {TEXT -1 1 "\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "isprime((2^128 + 2^32 + 1)/( 3*3*3*3*80271439));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ifac tor((2^128 + 2^32 + 1)/(3*3*3*3*80271439), easy);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 64 "gen_P_F((2^128 + 2^32 + 1)/(3*3*3*3*80271439 ), x -> x^2 + 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "isp rime((2^128 + 2^32 + 1)/(3*3*3*3*80271439*995910073));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}}{MARK "99" 0 }{VIEWOPTS 1 1 0 1 1 1803 }