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Below I present - with proof - the ( quite fortuitous) discovery " }}{PARA 0 "" 0 "" {TEXT -1 6 " of a " } {TEXT 326 16 "2000-digit prime" }{TEXT -1 29 ". I would like to call \+ it a " }{TEXT 327 16 "millennium prime" }{TEXT -1 2 ".]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 12 "Introduction" } {TEXT -1 55 ". In a separate Maple worksheet (1914_pap.mws) I have " }}{PARA 0 "" 0 "" {TEXT -1 39 "worked through H.C.Pocklington's paper \+ " }{TEXT 256 88 "The Determination of the \nPrime or Composite Nature \+ of Large Numbers by Fermat's theorem" }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 27 "which was published in the " }{TEXT 257 43 "Proceedings of the Cambridge Philosophical " }}{PARA 0 "" 0 "" {TEXT 266 7 "Socie ty" }{TEXT -1 65 " in 1916 (it was only two pages long!), and \"Read\" by Pocklington" }}{PARA 0 "" 0 "" {TEXT -1 72 "on the 9th. of March 1 914. In modern expositions of Pocklington's paper" }}{PARA 0 "" 0 "" {TEXT -1 45 "there is some lack of consistency as to what " }{TEXT 267 2 "is" }{TEXT -1 2 " '" }{TEXT 268 21 "Pocklington's Theorem" } {TEXT -1 1 "'" }}{PARA 0 "" 0 "" {TEXT -1 68 "- possibly because at no point in his paper does Pocklington himself" }}{PARA 0 "" 0 "" {TEXT -1 72 "actually state a theorem! - one has to weed it (actually them) \+ out from " }}{PARA 0 "" 0 "" {TEXT -1 75 "his (beautiful) analysis of \+ some consequences of Fermat's 'little' theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "What I did in 1914_pap.mw s was to go back to Pocklington's original paper," }}{PARA 0 "" 0 "" {TEXT -1 70 "work through his presentation (changing some notation her e and there -" }}{PARA 0 "" 0 "" {TEXT -1 24 "I don't like his using ' " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 51 "' as a symbol for a pri me - and also try to make it" }}{PARA 0 "" 0 "" {TEXT -1 71 "more unde rstandable), and weave in some illustrative Maple computations" }} {PARA 0 "" 0 "" {TEXT -1 72 "(Pocklington's paper does not contain a s ingle numerical illustration!)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 73 "My aim in this worksheet was to present s ome illustrations of his theorem" }}{PARA 0 "" 0 "" {TEXT -1 79 " - or rather one version of one of his theorems - to some specially constru cted" }}{PARA 0 "" 0 "" {TEXT -1 76 "numbers, to present to my third y ear B.Ed. and B.A. students in this year's " }{TEXT 331 32 "\nNumber T heory and Cryptography " }{TEXT -1 7 "course." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Anyone well versed in num ber theoretical arts will " }{TEXT 330 0 "" }{TEXT -1 21 "immediately \+ recognise" }}{PARA 0 "" 0 "" {TEXT -1 34 "that the numbers I have chos en as " }{TEXT 270 19 "possible candidates" }{TEXT -1 23 " for being p rime stood " }}{PARA 0 "" 0 "" {TEXT 269 24 "little chance of success " }{TEXT 271 0 "" }{TEXT -1 54 "; rather they were chosen (from a huge range of other " }}{PARA 0 "" 0 "" {TEXT -1 76 "equally interesting p ossible ones) because of the beauty of their structure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "I have (like any num ber theorist) a deep personal attachment to the " }{TEXT 272 10 "Eucli dean " }}{PARA 0 "" 0 "" {TEXT 275 7 "numbers" }{TEXT -1 27 " - the nu mbers of the form " }{XPPEDIT 18 0 "p[1]*p[2]*p[3];" "6#*(&%\"pG6#\"\" \"\"\"\"&F%6#\"\"#F(&F%6#\"\"$F(" }{TEXT -1 5 " ... " }{XPPEDIT 18 0 " p[n-1]*p[n]+1;" "6#,&*&&%\"pG6#,&%\"nG\"\"\"\"\"\"!\"\"F*&F&6#F)F*F*\" \"\"F*" }{TEXT -1 9 " - where " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "p[1], p[2],p[3];" "6%&%\"pG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 8 ", ... \+ , " }{XPPEDIT 18 0 "p[n-1],p[n];" "6$&%\"pG6#,&%\"nG\"\"\"\"\"\"!\"\"& F$6#F'" }{TEXT -1 15 " are the first " }{TEXT 273 1 "n" }{TEXT -1 36 " prime numbers, and what I have done" }}{PARA 0 "" 0 "" {TEXT -1 19 "h ere was to choose " }{XPPEDIT 18 0 "p[n+1];" "6#&%\"pG6#,&%\"nG\"\"\" \"\"\"F(" }{TEXT -1 15 ", and form its " }{TEXT 274 1 "n" }{TEXT -1 10 "-th power " }{XPPEDIT 18 0 "p[n+1]^n;" "6#)&%\"pG6#,&%\"nG\"\"\"\" \"\"F)F(" }{TEXT -1 20 ". That latter number" }}{PARA 0 "" 0 "" {TEXT -1 34 "- which is obviously greater than " }{XPPEDIT 18 0 "p[1]*p[2]*p [3];" "6#*(&%\"pG6#\"\"\"\"\"\"&F%6#\"\"#F(&F%6#\"\"$F(" }{TEXT -1 5 " ... " }{XPPEDIT 18 0 "p[n-1]*p[n];" "6#*&&%\"pG6#,&%\"nG\"\"\"\"\"\"! \"\"F)&F%6#F(F)" }{TEXT -1 11 " - is the '" }{XPPEDIT 18 0 "p^alpha;" "6#)%\"pG%&alphaG" }{TEXT -1 8 "' in the" }}{PARA 0 "" 0 "" {TEXT -1 36 "following theorem, as is the number " }{XPPEDIT 18 0 "p[1]*p[2]*p[ 3];" "6#*(&%\"pG6#\"\"\"\"\"\"&F%6#\"\"#F(&F%6#\"\"$F(" }{TEXT -1 5 " \+ ... " }{XPPEDIT 18 0 "p[n-1]*p[n];" "6#*&&%\"pG6#,&%\"nG\"\"\"\"\"\"! \"\"F)&F%6#F(F)" }{TEXT -1 5 " the " }{TEXT 276 2 "'U" }{TEXT -1 8 "' \+ of it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 258 2 "A " }{TEXT -1 0 "" }{TEXT 259 19 "Pocklington theorem " }{TEXT -1 7 ": Let " }{TEXT 277 1 "N" }{TEXT -1 29 " be a natural n umber and let " }{XPPEDIT 18 0 "N-1 = p^alpha*U;" "6#/,&%\"NG\"\"\"\" \"\"!\"\"*&)%\"pG%&alphaGF&%\"UGF&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 6 "where " }{TEXT 278 1 "p" }{TEXT -1 11 " is prime, " } {XPPEDIT 18 0 "p^alpha;" "6#)%\"pG%&alphaG" }{TEXT -1 25 " is the larg est power of " }{TEXT 279 1 "p" }{TEXT -1 11 " dividing (" }{XPPEDIT 18 0 "N-1;" "6#,&%\"NG\"\"\"\"\"\"!\"\"" }{TEXT -1 7 "), and " } {XPPEDIT 18 0 "U < p^alpha;" "6#2%\"UG)%\"pG%&alphaG" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 " \+ Suppose also that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "a^(N-1) = 1;" "6#/)%\"aG,&%\"NG\"\"\"\"\"\"!\"\"\"\"\"" }{TEXT -1 6 " (mod " }{TEXT 281 1 "N" }{TEXT -1 19 ") for some integer " }{TEXT 257 1 "a" }{TEXT -1 13 " ... (" }{TEXT 287 1 "i" }{TEXT -1 57 " ) \n\n and " }}{PARA 15 "" 0 "" {TEXT 260 2 " " }{XPPEDIT 18 0 "gcd(a^((N-1)/p)-1,N) = 1; " "6#/-%$gcdG6$,&)%\"aG*&,&%\"NG\"\"\"\"\"\"!\"\"F-%\"pGF/F-\"\"\"F/F, \"\"\"" }{TEXT -1 35 " ... (" }{TEXT 288 2 "ii" }{TEXT -1 2 ")\n" }}{PARA 0 "" 0 "" {TEXT 261 0 "" }{TEXT 262 5 "then " }{TEXT 263 1 "N" }{TEXT 280 9 " is prime" }{TEXT -1 3 ". [" }{TEXT 283 3 "end" }{TEXT -1 1 "]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 65 "So, I now wish to apply that theorem to t he sequence of numbers \{" }{XPPEDIT 18 0 "f[n];" "6#&%\"fG6#%\"nG" } {TEXT -1 2 "\}," }}{PARA 0 "" 0 "" {TEXT -1 6 "where:" }}{PARA 0 "" 0 "" {TEXT -1 38 " " }{XPPEDIT 18 0 "f[1];" "6#&%\"fG6#\"\"\"" }{TEXT -1 11 " = 2*3 + 1," }}{PARA 0 "" 0 "" {TEXT -1 37 " " }{XPPEDIT 18 0 "f[2];" "6#&%\"fG6#\"\"#" }{TEXT -1 7 " = 2*3*" }{XPPEDIT 18 0 "5^2; " "6#*$\"\"&\"\"#" }{TEXT -1 5 " + 1," }}{PARA 0 "" 0 "" {TEXT -1 37 " " }{XPPEDIT 18 0 "f[3];" "6#&%\"f G6#\"\"$" }{TEXT -1 7 " = 2*3*" }{XPPEDIT 18 0 "5;" "6#\"\"&" }{TEXT -1 1 "*" }{XPPEDIT 18 0 "7^3;" "6#*$\"\"(\"\"$" }{TEXT -1 5 " + 1," }} {PARA 0 "" 0 "" {TEXT -1 43 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 43 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 37 " \+ " }{XPPEDIT 18 0 "f[n];" "6#&%\"fG6#%\"nG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "p[1]*p[2]*p[3];" "6#*(&%\"pG6#\"\"\"\"\"\"&F%6#\"\"#F(& F%6#\"\"$F(" }{TEXT -1 5 " ... " }{XPPEDIT 18 0 "p[n-1]*p[n]*p[n+1]^n+ 1;" "6#,&*(&%\"pG6#,&%\"nG\"\"\"\"\"\"!\"\"F*&F&6#F)F*)&F&6#,&F)F*\"\" \"F*F)F*F*\"\"\"F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 310 12 "Nomenclature" }{TEXT -1 4 ". [\"" }{TEXT 311 22 "What's in a name? ... " }{TEXT -1 27 "\"] I w ould like to call the" }}{PARA 0 "" 0 "" {TEXT -1 14 "above numbers " }{TEXT 312 1 "'" }{TEXT 313 26 "L shaped Euclidean numbers" }{TEXT -1 17 "', the name being" }}{PARA 0 "" 0 "" {TEXT -1 35 "suggested by the following picture:" }}{PARA 0 "" 0 "" {TEXT -1 65 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 145 " \+ 7\n \+ *" }}{PARA 0 "" 0 "" {TEXT -1 72 " \+ 7" }}{PARA 0 "" 0 "" {TEXT -1 72 " \+ *" }}{PARA 0 "" 0 "" {TEXT -1 89 " \+ 7 \+ " }}{PARA 0 "" 0 "" {TEXT -1 53 " \+ " }{XPPEDIT 18 0 "f[3];" "6#&%\"fG6#\"\"$" }{TEXT -1 13 " = 2*3*5* + 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 285 49 "The L shaped Euclidean numbers can be computed by" }} {PARA 0 "" 0 "" {TEXT 332 23 "defining the function '" }{TEXT 284 1 "f " }{TEXT 286 13 " ' as follows" }{TEXT -1 26 ": \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f := n -> product(ith prime(k), k=1..n)*ithprime(n+1)^n + 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"nG6\"6$%)operatorG%&arrowGF(,&*&-%(productG6$-%)ithp rimeG6#%\"kG/F4;\"\"\"9$F7)-F26#,&F8F7F7F7F8F7F7F7F7F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(1); # 2*3 + 1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " f(2); # 2*3*5^2 + 1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$^\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f(3); # 2*3*5*7^3 + 1:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"&\"H5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "These - and o thers - are the numbers that I wish to subject to the " }}{PARA 0 "" 0 "" {TEXT -1 48 "above Pocklington theorem to test for primality." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Before la unching into a systematic search, I will illustrate " }{TEXT 333 3 "tw o" }{TEXT -1 11 " individual" }}{PARA 0 "" 0 "" {TEXT 334 8 "examples " }{TEXT -1 4 " - " }{XPPEDIT 18 0 "f[9];" "6#&%\"fG6#\"\"*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f[10];" "6#&%\"fG6#\"#5" }{TEXT -1 8 " - \+ with " }{TEXT 292 0 "" }{TEXT -1 1 "'" }{TEXT 293 1 "a" }{TEXT -1 17 " ' chosen to be 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"7JS&fcl7$3WOK" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "2&^(f(9) - 1) mod f(9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"7(**>,(odk*fP1$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Thus " } {XPPEDIT 18 0 "f[9];" "6#&%\"fG6#\"\"*" }{TEXT -1 23 " has failed cond ition (" }{TEXT 289 1 "i" }{TEXT -1 32 ") of Pocklington - which, bein g " }{TEXT 290 8 "Fermat's" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 291 18 "test to the base 2" }{TEXT -1 26 " - means immediately that " }{XPPEDIT 18 0 "f[9];" "6#&%\"fG6#\"\"*" }{TEXT -1 14 " is composite. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Now t o test " }{XPPEDIT 18 0 "f[10];" "6#&%\"fG6#\"#5" }{TEXT -1 1 ":" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\":Jsnp`='RzNu-`" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "2&^(f(10) - 1) mod f(10); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "igcd(2&^((f(10)-1 )/ithprime(10)) mod f(10) - 1, f(10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 "f[10];" "6#&%\"fG6# \"#5" }{TEXT -1 4 " is " }{TEXT 294 6 "proved" }{TEXT -1 7 " to be " } {TEXT 335 5 "prime" }{TEXT -1 26 " by Pocklington's theorem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Now to engage i n a systematic search, " }{TEXT 297 16 "in which I fix '" }{TEXT 296 1 "a" }{TEXT 298 6 "' at 2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "I " }{TEXT 302 5 "first" }{TEXT -1 28 " seek numbers which satisfy " }{TEXT 306 11 "condition (" } {TEXT 295 1 "i" }{TEXT 307 1 ")" }{TEXT -1 18 " of Pocklington's " }} {PARA 0 "" 0 "" {TEXT -1 46 "theorem; that is, I simply look for value s of " }{TEXT 299 0 "" }{TEXT -1 1 "'" }{TEXT 300 1 "k" }{TEXT -1 13 " ' for which " }{XPPEDIT 18 0 "f[k];" "6#&%\"fG6#%\"kG" }{TEXT -1 1 " \+ " }{TEXT 301 35 "passes\nFermat's test to the base 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 12 "Subsequently" }{TEXT -1 43 " I will further subject the corresponding " }{TEXT 303 1 "f" } {TEXT -1 7 " 's to " }}{PARA 0 "" 0 "" {TEXT 308 11 "condition (" } {TEXT 304 2 "ii" }{TEXT 309 1 ")" }{TEXT -1 47 " of Pocklington's theo rem. [Of course I realise" }}{PARA 0 "" 0 "" {TEXT -1 56 "that I could have incorporated the two conditions into a" }}{PARA 0 "" 0 "" {TEXT -1 18 "single procedure.]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "candidates := proc(m, n)\nlocal k;\nfor k from m to n do\nif 2&^( f(k)-1) mod f(k) = 1\nthen print(k)\nfi\nod\nend:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "candidates(1, 30);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#?" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "So, we have four candidates out of the first thirty L shaped" }}{PARA 0 "" 0 "" {TEXT -1 18 "Euclidean numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The first two produce primes (" } {XPPEDIT 18 0 "f[1];" "6#&%\"fG6#\"\"\"" }{TEXT -1 20 " = 2*3 + 1 = 7, and " }{XPPEDIT 18 0 "f[2];" "6#&%\"fG6#\"\"#" }{TEXT -1 7 " = 2*3*" }{XPPEDIT 18 0 "5^2;" "6#*$\"\"&\"\"#" }{TEXT -1 12 " + 1 = 151, " }} {PARA 0 "" 0 "" {TEXT -1 33 "are prime). I have already shown " } {XPPEDIT 18 0 "f[10];" "6#&%\"fG6#\"#5" }{TEXT -1 29 " is prime above, and so only " }{XPPEDIT 18 0 "f[20];" "6#&%\"fG6#\"#?" }{TEXT -1 9 " \+ remains:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "igcd(2&^((f(20) -1)/ithprime(20)) mod f(20) - 1, f(20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "So " }{XPPEDIT 18 0 "f[20];" "6#&%\"fG6#\" #?" }{TEXT -1 31 " is also prime, and here it is:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "f(20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \\o\"RBUD%>M#*e(3\"Q4#4!o.m$R<6.Rw.$oLr$\\I5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "length(f(20));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#l" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "I now proceed to search for others:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "candidates(31, 60); # no output" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "candidates(61, 120); # no o utput" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "candidates(121, 18 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$t\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "igcd(2&^((f(173)-1)/ithprime(173)) mod f(173) - \+ 1, f(173));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "length(f(173));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"$_*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Great!!! It follows that " } {XPPEDIT 18 0 "f[173];" "6#&%\"fG6#\"$t\"" }{TEXT -1 34 " is prime, an d it has 952 digits!!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "I want a '" }{TEXT 264 7 "titanic" }{TEXT -1 50 "' pri me (one with at least 1000 digits) though!! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "On we go:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "candidates(181, 240); # no output " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "candidates(241, 300); \+ # no output" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "length(f(300 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%B=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "candidates(301, 330);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$D$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 336 12 "At long last" }{TEXT -1 31 " - a nother L shaped candidate!!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Let's subject it to condition (" }{TEXT 314 2 " ii" }{TEXT -1 27 ") of Pocklingtho's theorem:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "igcd(2&^((f(325)-1)/ithprime(325)) mod f(325) - \+ 1, f(325));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Great!!! " }{XPPEDIT 18 0 "f[325];" "6#&%\"fG6#\"$D$" }{TEXT -1 48 " is prime!!! And how many digits does it have?:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "length(f(325));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"%+?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Wow!!! I'd like to call it a \+ " }{TEXT 315 9 "millenium" }{TEXT 337 6 " prime" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "And here \+ it is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(325);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"[hr\"pS\"3Al4xxF\"[x\"y(\\qfI4&GY?/OfC(3?# ygY$)z._MG!fXn!fg!\\'=yTy1U*\\Qw1Pq_\">#*o]0!z_*\\8z/-'z?V_3&eb;$oU(R' )Qfv7M^pXxFDIDOk$Q'GsxfjYt%>l^tKj_\"*p,f/;G'zUD1\"QA;^'Qi7t8%R33hT+egZ 8P\\vzUgIZ?2F'e4sKpjm%epO[/%>2m-&*ehk/$RC!)e)y#H!Q(z[lct&Rc&R6XJJ=0!** 4\\&)44OYK8dWHATvn0$3/Hu)zEMR8m$y>*H>Yk<`B>B\"GwDj[spE%>w$G$pE!fftR/-N 0az%H>d;j*y/?2w=x\"=nHb*=(**G/\")Ha,d9aeV;#H6m%*yFX)4*pM;V]l# G)z2!f[Y*)eM3=A(enASY`@Vd0<.1UU)z9OS7rS0dF7([]!*zBp6lx:(R#*=(pGHao%GtK wI-V_!)QtCe^'p\\#3=Vn;b'opd<%H%*47A?a`\\iy0G![nJHu![(H\"4@MgpwJ.oDke!Q Bbyi]?#y2d+lor#R.L(*zHRh[;=S!>(y\"3Qy&G)[A$zV\\ZCJT%f8Y\"*>k[mSI[2CMM \\'eo30Kc`M')o)>y4No#G\"*)*Q%>qrHfasp<**=J<[[PXQ_h_a]_yc'o\"\\#evyYc#R %*)Gp;E5$\\Ob*fcAEn/DpU*36,_?['pbuqg`8B,hXSlNH8?P\"[$*oD'GK!=`a T!3dv<4m\"fEI&pN\"Q@=&)3zI'p%3.*=8')>T77T+[9>i9#4x8[sDyp6za&yN\"*z?D.* 3'R$>Vg2\"[&pB0+?7NW`J<)>i'>ZoQO2(ym\">W&o(fgH=s))eH6\\0f&))fF$Q)y$R*))4G\"p9\"*HNb4#fg vw/,1Xh\"ysW'*Gw;Q#R%>gF&4+e/f[wn(=lw,:RWWRYvAGSe_@ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 317 11 "This is how" }{TEXT -1 1 " " }{XPPEDIT 18 0 "f[325];" "6#&%\"fG6#\"$D$" }{TEXT -1 1 " " }{TEXT 319 14 "is con structed" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 316 5 "First" }{TEXT -1 28 ", form the first 325 primes: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(ithprime(k), k=1..3 25);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6a_l\"\"#\"\"$\"\"&\"\"(\"#6\"#8 \"#<\"#>\"#B\"#H\"#J\"#P\"#T\"#V\"#Z\"#`\"#f\"#h\"#n\"#r\"#t\"#z\"#$) \"#*)\"#(*\"$,\"\"$.\"\"$2\"\"$4\"\"$8\"\"$F\"\"$J\"\"$P\"\"$R\"\"$\\ \"\"$^\"\"$d\"\"$j\"\"$n\"\"$t\"\"$z\"\"$\"=\"$\">\"$$>\"$(>\"$*>\"$6# \"$B#\"$F#\"$H#\"$L#\"$R#\"$T#\"$^#\"$d#\"$j#\"$p#\"$r#\"$x#\"$\"G\"$$ G\"$$H\"$2$\"$6$\"$8$\"$<$\"$J$\"$P$\"$Z$\"$\\$\"$`$\"$f$\"$n$\"$t$\"$ z$\"$$Q\"$*Q\"$(R\"$,%\"$4%\"$>%\"$@%\"$J%\"$L%\"$R%\"$V%\"$\\%\"$d%\" $h%\"$j%\"$n%\"$z%\"$([\"$\"\\\"$*\\\"$.&\"$4&\"$@&\"$B&\"$T&\"$Z&\"$d &\"$j&\"$p&\"$r&\"$x&\"$(e\"$$f\"$*f\"$,'\"$2'\"$8'\"$<'\"$>'\"$J'\"$T '\"$V'\"$Z'\"$`'\"$f'\"$h'\"$t'\"$x'\"$$o\"$\"p\"$,(\"$4(\"$>(\"$F(\"$ L(\"$R(\"$V(\"$^(\"$d(\"$h(\"$p(\"$t(\"$(y\"$(z\"$4)\"$6)\"$@)\"$B)\"$ F)\"$H)\"$R)\"$`)\"$d)\"$f)\"$j)\"$x)\"$\"))\"$$))\"$())\"$2*\"$6*\"$> *\"$H*\"$P*\"$T*\"$Z*\"$`*\"$n*\"$r*\"$x*\"$$)*\"$\"**\"$(**\"%45\"%85 \"%>5\"%@5\"%J5\"%L5\"%R5\"%\\5\"%^5\"%h5\"%j5\"%p5\"%(3\"\"%\"4\"\"%$ 4\"\"%(4\"\"%.6\"%46\"%<6\"%B6\"%H6\"%^6\"%`6\"%j6\"%r6\"%\"=\"\"%(=\" \"%$>\"\"%,7\"%87\"%<7\"%B7\"%H7\"%J7\"%P7\"%\\7\"%f7\"%x7\"%z7\"%$G\" \"%*G\"\"%\"H\"\"%(H\"\"%,8\"%.8\"%28\"%>8\"%@8\"%F8\"%h8\"%n8\"%t8\"% \"Q\"\"%*R\"\"%49\"%B9\"%F9\"%H9\"%L9\"%R9\"%Z9\"%^9\"%`9\"%f9\"%r9\"% \"[\"\"%$[\"\"%([\"\"%*[\"\"%$\\\"\"%*\\\"\"%6:\"%B:\"%J:\"%V:\"%\\:\" %`:\"%f:\"%n:\"%r:\"%z:\"%$e\"\"%(f\"\"%,;\"%2;\"%4;\"%8;\"%>;\"%@;\"% F;\"%P;\"%d;\"%j;\"%n;\"%p;\"%$p\"\"%(p\"\"%*p\"\"%4<\"%@<\"%B<\"%L<\" %T<\"%Z<\"%`<\"%f<\"%x<\"%$y\"\"%(y\"\"%*y\"\"%,=\"%6=\"%B=\"%J=\"%Z= \"%h=\"%n=\"%r=\"%t=\"%x=\"%z=\"%*)=\"%,>\"%2>\"%8>\"%J>\"%L>\"%\\>\"% ^>\"%t>\"%z>\"%()>\"%$*>\"%(*>\"%**>\"%.?\"%6?\"% " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 318 4 "Next" }{TEXT -1 21 ", form their product: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "U := product(ithprime(k ), k=1..325);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"UG\"_dn!pg\"Q:?\" H'=@]gH*o.Zr!fMb'*R3dUho*R)[!3p7cm]7CeAN\">d[F>S^L!y@\"o!3N7a$oqx@ma!)p8u>;d!pD'GMw\\A\\jVwiKt/['p#G\\:,JWX fwL0])G)pI=C9))**QD%yHr]L]*RN))pfS6k/av)*\\]+z Zj\"o`-^RxG!*)oE%35-(HKd$3U3034J\\Cyr&)[@D4\"Q'=S9$zC@!f')*Q]AJ9nx.1?Kgb*>wA+wh[)=D!Hd2Qn0O: Z'y]CyS4#=J1ztk$pnjm1i,I3*)G`&o3byIY]V$Qmzl@$RH![HvYL*e>;8V$[X3xE--I*z uqf3I,:jETiDPncRxWBS@sp6]3!QmVT&eq\"=cGleK#Q1M*=(H)G;c]4.;D0i^q]RlB(>+ 8xLtv1+p^'*G!=M&)=DIsFH" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 338 3 "Now" }{TEXT -1 7 ", let '" } {TEXT 320 1 "p" }{TEXT -1 30 "' be the 326-th. prime number:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p := ithprime(326);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"%h@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 339 4 "Form" } {TEXT -1 33 " the 325-th. power of that prime " }{TEXT 321 1 "p" } {TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p^325;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#\"g^o,?%4dRzz+L3N1.XA^p?ytzBm(oLHv_[C3 F5'p[C&3Z)e(HJg/S/Wvx3ac!f[Q7%oas8;8eh#oM$R'\\YUi@$)Q>P.O?)o`3([ihu(zE 75/hSk$Q)[rv+Hm)))*z]p3e^p7N%z9[,X.w1,h#R\\8h1=+Q7#[x7FZ$*pGxq\\r)[7![ (Q$3:m&GkHT!y@n.fPv)HInwd4V[W\\=/V$38+9uoah9(Q*G\")f\\W(ot!=$z.SK(GPPQ o]`506if3%[I)4+GUaX,'o%=yqi)p#\\k=(z-#*p+\\!*f(3\"*=;l9R9R8x(GC\\`[X$o @Lh'*y>4)[Bm%f7FLNEhz-QjK\\&3C<$pA4WXfvx6[QWdeNY52 e.Z]jha])Qtvm-F%=Iy8_0mTO!4;z&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 322 7 "Finally" }{TEXT -1 18 ", form the number " }{TEXT 323 1 "N" }{TEXT -1 13 ", defined by " }{XPPEDIT 18 0 "N = p^325*U+1;" "6#/%\"NG,&*&%\"pG\"$D$%\"UG\"\"\"F*\"\"\"F*" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "That " }{TEXT 324 1 "N" }{TEXT -1 32 " is the number 2000- digit prime " }{XPPEDIT 18 0 "f[325];" "6#&%\"fG6#\"$D$" }{TEXT -1 23 ", which I would like to" }}{PARA 0 "" 0 "" {TEXT -1 6 "call '" } {TEXT 325 16 "millennium prime" }{TEXT -1 15 "'. Here it is:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "millennium_prime := p^325*U + 1;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%1millennium_primeG\"[hr\"pS\"3Al4xxF\"[x\"y(\\qfI4&GY?/OfC(3?# ygY$)z._MG!fXn!fg!\\'=yTy1U*\\Qw1Pq_\">#*o]0!z_*\\8z/-'z?V_3&eb;$oU(R' )Qfv7M^pXxFDIDOk$Q'GsxfjYt%>l^tKj_\"*p,f/;G'zUD1\"QA;^'Qi7t8%R33hT+egZ 8P\\vzUgIZ?2F'e4sKpjm%epO[/%>2m-&*ehk/$RC!)e)y#H!Q(z[lct&Rc&R6XJJ=0!** 4\\&)44OYK8dWHATvn0$3/Hu)zEMR8m$y>*H>Yk<`B>B\"GwDj[spE%>w$G$pE!fftR/-N 0az%H>d;j*y/?2w=x\"=nHb*=(**G/\")Ha,d9aeV;#H6m%*yFX)4*pM;V]l# G)z2!f[Y*)eM3=A(enASY`@Vd0<.1UU)z9OS7rS0dF7([]!*zBp6lx:(R#*=(pGHao%GtK wI-V_!)QtCe^'p\\#3=Vn;b'opd<%H%*47A?a`\\iy0G![nJHu![(H\"4@MgpwJ.oDke!Q Bbyi]?#y2d+lor#R.L(*zHRh[;=S!>(y\"3Qy&G)[A$zV\\ZCJT%f8Y\"*>k[mSI[2CMM \\'eo30Kc`M')o)>y4No#G\"*)*Q%>qrHfasp<**=J<[[PXQ_h_a]_yc'o\"\\#evyYc#R %*)Gp;E5$\\Ob*fcAEn/DpU*36,_?['pbuqg`8B,hXSlNH8?P\"[$*oD'GK!=`a T!3dv<4m\"fEI&pN\"Q@=&)3zI'p%3.*=8')>T77T+[9>i9#4x8[sDyp6za&yN\"*z?D.* 3'R$>Vg2\"[&pB0+?7NW`J<)>i'>ZoQO2(ym\">W&o(fgH=s))eH6\\0f&))fF$Q)y$R*))4G\"p9\"*HNb4#fg vw/,1Xh\"ysW'*Gw;Q#R%>gF&4+e/f[wn(=lw,:RWWRYvAGSe_@ " 0 "" {MPLTEXT 1 0 25 "length(millenni um_prime);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%+?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 3 4" 1 }{VIEWOPTS 1 1 0 1 1 1803 }