# perl6/perl6-examples

some new exercises and a pod version of the 99 problems document

893 99-problems/99-problems.pod
115 99-problems/P36-ovid.pl
 @@ -0,0 +1,115 @@ +use v6; + +# Specification: +# P36 (**) Determine the prime factors of a given positive integer (2). +# Construct a list containing the prime factors and their multiplicity. +# Example: +# > prime_factors_mult(315).perl.say +# (3 => 2, 5 => 1, 7 => 1) +# +# Hint: The problem is similar to problem P13. + + +# This was originally a blog post: +# http://blogs.perl.org/users/ovid/2010/08/prime-factors-in-perl-6.html + +# first some boring auxiliary stuff + +my @PRIMES = <2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 +59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 +167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 +271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 +389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 +503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 +631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 +757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 +883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 +1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 +1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 +1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 +1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 +1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 +1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 +1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 +1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 1879 1889 1901 +1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997 1999 2003 2011 2017 +2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 +2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 +2269 2273 2281 2287 2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 +2377 2381 2383 2389 2393 2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 +2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617 2621 +2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2693 2699 2707 2711 2713 +2719 2729 2731 2741 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 +2837 2843 2851 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 +2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 +3089 3109 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 +3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 +3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 3463 3467 +3469 3491 3499 3511 3517 3527 3529 3533 3539 3541 3547 3557 3559 3571 3581 +3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671 3673 3677 3691 3697 +3701 3709 3719 3727 3733 3739 3761 3767 3769 3779 3793 3797 3803 3821 3823 +3833 3847 3851 3853 3863 3877 3881 3889 3907 3911 3917 3919 3923 3929 3931 +3943 3947 3967 3989 4001 4003 4007 4013 4019 4021 4027 4049 4051 4057 4073 +4079 4091 4093 4099 4111 4127 4129 4133 4139 4153 4157 4159 4177 4201 4211 +4217 4219 4229 4231 4241 4243 4253 4259 4261 4271 4273 4283 4289 4297 4327 +4337 4339 4349 4357 4363 4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 +4463 4481 4483 4493 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 +4597 4603 4621 4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 +4723 4729 4733 4751 4759 4783 4787 4789 4793 4799 4801 4813 4817 4831 4861 +4871 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 +4987 4993 4999 5003 5009 5011 5021 5023 5039 5051 5059 5077 5081 5087 5099 +5101 5107 5113 5119 5147 5153 5167 5171 5179 5189 5197 5209 5227 5231 5233 +5237 5261 5273 5279 5281 5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 +5399 5407 5413 5417 5419 5431 5437 5441 5443 5449 5471 5477 5479 5483 5501 +5503 5507 5519 5521 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639 5641 +5647 5651 5653 5657 5659 5669 5683 5689 5693 5701 5711 5717 5737 5741 5743 +5749 5779 5783 5791 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861 +5867 5869 5879 5881 5897 5903 5923 5927 5939 5953 5981 5987 6007 6011 6029 +6037 6043 6047 6053 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133 6143 +6151 6163 6173 6197 6199 6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 +6277 6287 6299 6301 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367 6373 +6379 6389 6397 6421 6427 6449 6451 6469 6473 6481 6491 6521 6529 6547 6551 +6553 6563 6569 6571 6577 6581 6599 6607 6619 6637 6653 6659 6661 6673 6679 +6689 6691 6701 6703 6709 6719 6733 6737 6761 6763 6779 6781 6791 6793 6803 +6823 6827 6829 6833 6841 6857 6863 6869 6871 6883 6899 6907 6911 6917 6947 +6949 6959 6961 6967 6971 6977 6983 6991 6997 7001 7013 7019 7027 7039 7043 +7057 7069 7079 7103 7109 7121 7127 7129 7151 7159 7177 7187 7193 7207 7211 +7213 7219 7229 7237 7243 7247 7253 7283 7297 7307 7309 7321 7331 7333 7349 +7351 7369 7393 7411 7417 7433 7451 7457 7459 7477 7481 7487 7489 7499 7507 +7517 7523 7529 7537 7541 7547 7549 7559 7561 7573 7577 7583 7589 7591 7603 +7607 7621 7639 7643 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723 7727 +7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 +7883 7901 7907 7919>; +my %IS-PRIME = map {; \$_ => 1 }, @PRIMES; + +sub is-prime(Int \$number) { + return False if \$number < 2; # special case + die "\$number is too large" if \$number > @PRIMES[*-1]**2; + return %IS-PRIME{\$number}; +} + + +# here is the main function +sub prime-factors(Int \$number is copy) { + # don't try to factor prime numbers + return { \$number => 1 } if is-prime(\$number); + + my %factors; + for @PRIMES -> \$prime-number { + last if \$prime-number ** 2 > \$number; + while \$number % \$prime-number == 0 { + %factors{\$prime-number} //= 0; + %factors{\$prime-number}++; + \$number /= \$prime-number; + } + } + %factors{\$number}++ if \$number != 1; # we have a prime left over + return %factors; +} + +for 17, 53, 90, 94, 200, 289, 62710561 -> \$number { + say "Prime factors of \$number are: ", prime-factors(\$number).perl; +} + + +# vim:ft=perl6
6 99-problems/P36-rhebus.pl
 @@ -5,7 +5,7 @@ # Construct a list containing the prime factors and their multiplicity. # Example: # > prime_factors_mult(315).perl.say -# ([3,2],[5,1],[7,1]) +# (3 => 2, 5 => 1, 7 => 1) # # Hint: The problem is similar to problem P13. @@ -18,13 +18,13 @@ (Int \$n) \$mult++; \$residue div= \$k; } - take [\$k, \$mult] if \$mult; + take \$k => \$mult if \$mult; last if \$residue == 1; # This if block is an optimisation which reduces number of iterations # for numbers with large prime factors (such as large primes) # It can be removed without affecting correctness. if \$k > sqrt \$residue { - take [\$residue,1]; + take \$residue => 1; last; } }
10 99-problems/P37-rhebus.pl
 @@ -22,10 +22,10 @@ (Int \$n) \$mult++; \$residue div= \$k; } - take [\$k, \$mult] if \$mult; + take \$k => \$mult if \$mult; last if \$residue == 1; if \$k > sqrt \$residue { - take [\$residue,1]; + take \$residue => 1; last; } } @@ -33,9 +33,9 @@ (Int \$n) # 1. One-liner version -say "phi(\$_): ", [*] prime_factors_mult(\$_).map({ (\$_[0]-1) * \$_[0] ** (\$_[1]-1) }) +say "phi(\$_): ", [*] prime_factors_mult(\$_).map({ (.key-1) * .key ** (.value-1) }) for 1..20; -say [*] prime_factors_mult(315).map: { (\$_[0]-1) * \$_[0] ** (\$_[1]-1) }; +say [*] prime_factors_mult(315).map: { (.key-1) * .key ** (.value-1) }; # 2. sub version @@ -45,7 +45,7 @@ (Int \$n) sub totient (Int \$n) { my @factors = prime_factors_mult(\$n); return [*] @factors.map: { - (\$_[0]-1) * \$_[0] ** (\$_[1]-1) + (.key-1) * .key ** (.value-1) } }
34 99-problems/P39-rhebus.pl
 @@ -0,0 +1,34 @@ +use v6; + +# Specification: +# P39 (*) A list of prime numbers. +# Given a range of integers by its lower and upper limit, construct a list +# of all prime numbers in that range. + + +# Copied from P31-rhebus.pl +sub is_prime (Int \$n) { + for 2..sqrt \$n -> \$k { + return Bool::False if \$n %% \$k; + } + return Bool::True; +} + + +# *@range is a slurpy parameter - it will swallow all the arguments passed +sub primes (*@range) { + gather for @range { + take \$_ if is_prime \$_; + } +} + +# we can call it with a range, as in the specification... +say ~primes(10..20); + +# or we can pass a list... +say ~primes(3,5,17,257,65537); + +# or a series... +say ~primes(1,2,*+*...100); + +# vim:ft=perl6
39 99-problems/P40-rhebus.pl
 @@ -0,0 +1,39 @@ +use v6; + +# Specification: +# P40 (**) Goldbach's conjecture. +# Goldbach's conjecture says that every positive even number greater +# than 2 is the sum of two prime numbers. Example: 28 = 5 + 23. It is +# one of the most famous facts in number theory that has not been proved +# to be correct in the general case. It has been numerically confirmed +# up to very large numbers (much larger than we can go with our Prolog +# system). Write a predicate to find the two prime numbers that sum up +# to a given even integer. +# +# Example: +# > say ~goldbach 28 +# 5 23 + + +# From P31-rhebus.pl again +sub is_prime (Int \$n) { + for 2..sqrt \$n -> \$k { + return Bool::False if \$n %% \$k; + } + return Bool::True; +} + + +# require even arguments +sub goldbach (Int \$n where {\$^a > 2 && \$^a %% 2}) { + for 2..\$n/2 -> \$k { + if is_prime(\$k) && is_prime(\$n-\$k) { + return (\$k, \$n-\$k); + } + } + return; # fail +} + +say ~goldbach \$_ for 28, 36, 52, 110; + +# vim:ft=perl6
62 99-problems/P41-rhebus.pl
 @@ -0,0 +1,62 @@ +use v6; + +# Specification: +# P41 (**) A list of Goldbach compositions. +# Given a range of integers by its lower and upper limit, print a list of all +# even numbers and their Goldbach composition. +# +# Example: +# > goldbach-list 9,20 +# 10 = 3 + 7 +# 12 = 5 + 7 +# 14 = 3 + 11 +# 16 = 3 + 13 +# 18 = 5 + 13 +# 20 = 3 + 17 +# +# In most cases, if an even number is written as the sum of two prime numbers, +# one of them is very small. Very rarely, the primes are both bigger than say +# 50. Try to find out how many such cases there are in the range 2..3000. +# +# Example (for a print limit of 50): +# > goldbach-list 1,2000,50 +# 992 = 73 + 919 +# 1382 = 61 + 1321 +# 1856 = 67 + 1789 +# 1928 = 61 + 1867 + + +# From P31-rhebus.pl again +sub is_prime (Int \$n) { + for 2..sqrt \$n -> \$k { + return Bool::False if \$n %% \$k; + } + return Bool::True; +} + + +# require even arguments +sub goldbach (Int \$n where {\$^a > 2 && \$^a %% 2}) { + for 2..\$n/2 -> \$k { + if is_prime(\$k) && is_prime(\$n-\$k) { + return (\$k, \$n-\$k); + } + } + # actually, it's more likely a logic error than a refutation :) + die "Goldbach's conjecture is false! \$n cannot be separated into two primes!" +} + +# Here we demonstrate an optional parameter with a default value +sub goldbach-list (Int \$low, Int \$high, Int \$limit = 1) { + for \$low .. \$high -> \$n { + next if \$n % 2; # skip invalid goldbach numbers + next if \$n == 2; + my @pair = goldbach(\$n); + say "\$n = ", @pair.join(' + ') if @pair[0] > \$limit; + } +} + +goldbach-list 9,20; +goldbach-list 2,3000,10; + +# vim:ft=perl6