Merge branch 'master' of git://github.com/killy971/factor
Slava Pestov
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USING: project-euler.065 tools.test ;
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IN: project-euler.065.tests
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[ 272 ] [ euler065 ] unit-test
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! Copyright (c) 2009 Guillaume Nargeot.
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! See http://factorcode.org/license.txt for BSD license.
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USING: kernel math lists lists.lazy project-euler.common sequences ;
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IN: project-euler.065
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! http://projecteuler.net/index.php?section=problems&id=065
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! DESCRIPTION
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! -----------
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! The square root of 2 can be written as an infinite continued fraction.
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! 1
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! √2 = 1 + -------------------------
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! 1
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! 2 + ---------------------
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! 1
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! 2 + -----------------
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! 1
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! 2 + -------------
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! 2 + ...
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! The infinite continued fraction can be written, √2 = [1;(2)], (2) indicates
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! that 2 repeats ad infinitum. In a similar way, √23 = [4;(1,3,1,8)].
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! It turns out that the sequence of partial values of continued fractions for
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! square roots provide the best rational approximations. Let us consider the
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! convergents for √2.
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! 1 3 1 7 1 17 1 41
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! 1 + - = - ; 1 + ----- = - ; 1 + --------- = -- ; 1 + ------------- = --
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! 2 2 1 5 1 12 1 29
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! 2 + - 2 + ----- 2 + ---------
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! 2 1 1
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! 2 + - 2 + -----
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! 2 1
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! 2 + -
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! 2
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! Hence the sequence of the first ten convergents for √2 are:
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! 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...
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! What is most surprising is that the important mathematical constant,
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! e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].
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! The first ten terms in the sequence of convergents for e are:
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! 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...
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! The sum of digits in the numerator of the 10th convergent is 1+4+5+7=17.
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! Find the sum of digits in the numerator of the 100th convergent of the
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! continued fraction for e.
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! SOLUTION
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! --------
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<PRIVATE
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: (e-frac) ( -- seq )
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2 lfrom [
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dup 3 mod zero? [ 3 / 2 * ] [ drop 1 ] if
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] lazy-map ;
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: e-frac ( n -- n )
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1 - (e-frac) ltake list>array reverse 0
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[ + recip ] reduce 2 + ;
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PRIVATE>
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: euler065 ( -- answer )
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100 e-frac numerator number>digits sum ;
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! [ euler065 ] 100 ave-time
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! 4 ms ave run time - 0.33 SD (100 trials)
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SOLUTION: euler065
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Guillaume Nargeot
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@ -30,7 +30,7 @@ IN: project-euler.072
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! The answer can be found by adding totient(n) for 2 ≤ n ≤ 1e6
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! The answer can be found by adding totient(n) for 2 ≤ n ≤ 1e6
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: euler072 ( -- answer )
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: euler072 ( -- answer )
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2 1000000 [a,b] [ totient ] [ + ] map-reduce ;
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2 1000000 [a,b] [ totient ] sigma ;
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! [ euler072 ] 100 ave-time
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! [ euler072 ] 100 ave-time
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! 5274 ms ave run time - 102.7 SD (100 trials)
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! 5274 ms ave run time - 102.7 SD (100 trials)
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Guillaume Nargeot
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Guillaume Nargeot
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Guillaume Nargeot
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Guillaume Nargeot
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Guillaume Nargeot
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Guillaume Nargeot
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USING: project-euler.188 tools.test ;
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IN: project-euler.188.tests
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[ 95962097 ] [ euler188 ] unit-test
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! Copyright (c) 2009 Guillaume Nargeot.
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! See http://factorcode.org/license.txt for BSD license.
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USING: kernel math math.functions project-euler.common ;
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IN: project-euler.188
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! http://projecteuler.net/index.php?section=problems&id=188
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! DESCRIPTION
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! -----------
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! The hyperexponentiation or tetration of a number a by a positive integer b,
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! denoted by a↑↑b or ^(b)a, is recursively defined by:
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! a↑↑1 = a,
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! a↑↑(k+1) = a^(a↑↑k).
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! Thus we have e.g. 3↑↑2 = 3^3 = 27, hence
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! 3↑↑3 = 3^27 = 7625597484987 and
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! 3↑↑4 is roughly 10^(3.6383346400240996*10^12).
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! Find the last 8 digits of 1777↑↑1855.
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! SOLUTION
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! --------
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! Using modular exponentiation.
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! http://en.wikipedia.org/wiki/Modular_exponentiation
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<PRIVATE
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: hyper-exp-mod ( a b m -- e )
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1 rot [ [ 2dup ] dip swap ^mod ] times 2nip ;
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PRIVATE>
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: euler188 ( -- answer )
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1777 1855 10 8 ^ hyper-exp-mod ;
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! [ euler188 ] 100 ave-time
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! 4 ms ave run time - 0.05 SD (100 trials)
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SOLUTION: euler188
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Guillaume Nargeot
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@ -16,15 +16,15 @@ USING: definitions io io.files io.pathnames kernel math math.parser
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project-euler.045 project-euler.046 project-euler.047 project-euler.048
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project-euler.045 project-euler.046 project-euler.047 project-euler.048
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project-euler.049 project-euler.051 project-euler.052 project-euler.053
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project-euler.049 project-euler.051 project-euler.052 project-euler.053
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project-euler.054 project-euler.055 project-euler.056 project-euler.057
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project-euler.054 project-euler.055 project-euler.056 project-euler.057
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project-euler.058 project-euler.059 project-euler.063 project-euler.067
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project-euler.058 project-euler.059 project-euler.063 project-euler.065
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project-euler.069 project-euler.071 project-euler.072 project-euler.073
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project-euler.067 project-euler.069 project-euler.071 project-euler.072
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project-euler.074 project-euler.075 project-euler.076 project-euler.079
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project-euler.073 project-euler.074 project-euler.075 project-euler.076
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project-euler.085 project-euler.092 project-euler.097 project-euler.099
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project-euler.079 project-euler.085 project-euler.092 project-euler.097
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project-euler.100 project-euler.102 project-euler.112 project-euler.116
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project-euler.099 project-euler.100 project-euler.102 project-euler.112
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project-euler.117 project-euler.124 project-euler.134 project-euler.148
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project-euler.116 project-euler.117 project-euler.124 project-euler.134
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project-euler.150 project-euler.151 project-euler.164 project-euler.169
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project-euler.148 project-euler.150 project-euler.151 project-euler.164
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project-euler.173 project-euler.175 project-euler.186 project-euler.190
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project-euler.169 project-euler.173 project-euler.175 project-euler.186
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project-euler.203 project-euler.215 ;
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project-euler.188 project-euler.190 project-euler.203 project-euler.215 ;
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IN: project-euler
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IN: project-euler
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<PRIVATE
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<PRIVATE
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