# publicperl6/roast

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 `use v6;use Test;plan 67;{    # P31 (**) Determine whether a given integer number is prime.    #     # Example:    # * (is-prime 7)    # T        # Very Naive implementation and     # could probably use something like:     # subset Divisible::Int of Int where { \$_ > 1 };    # sub is_prime(Divisible::Int \$num) {    # but "subset" is not working yet.        sub is_prime(Int \$num) returns Bool {                # 0 and 1 are not prime by definition        return Bool::False if \$num < 2;                # 2 and 3 are        return Bool::True if \$num < 4;            # no even number is prime        return Bool::False if \$num % 2 == 0;            # let's try what's left        my \$max=floor(sqrt(\$num));            # we could use        # for 3 ... *+2, \$max -> \$i {        # but it doesn't seem to work yet        loop (my \$i=3; \$i <= \$max ; \$i+=2) {            return Bool::False if \$num % \$i == 0;        }        return Bool::True;    }        ok !is_prime(0), "We should find that 0 is not prime";    ok !is_prime(1), ".. and neither is 1";    ok is_prime(2), ".. 2 is prime";    ok is_prime(3), ".. 3 is prime";    ok !is_prime(4), ".. 4 is not";    ok is_prime(5), ".. 5 is prime";    ok !is_prime(6), ".. 6 is even, thus not prime";    ok !is_prime(15), ".. 15 is product of two primes, but not prime";    ok is_prime(2531), ".. 2531 is a larger prime";    ok !is_prime(2533), ".. 2533 is not";}{    # P32 (**) Determine the greatest common divisor of two positive     # integer numbers.    #     # Use Euclid's algorithm.    # Example:    # * (gcd 36 63)    # 9        # Makes sense to declare types since gcd makes sense only for Ints.    # Yet, it should be possible to define it even for commutative rings    # other than Integers, so we use a multi sub.        our multi sub gcd(Int \$a, Int \$b){        return \$a if \$b == 0;        return gcd(\$b,\$a % \$b);    }    is gcd(36,63), 9, "We should be able to find the gcd of 36 and 63";    is gcd(63,36), 9, ".. and viceversa";    is gcd(0,5) , 5, '.. and that gcd(0,\$x) is \$x';    is gcd(0,0) , 0, '.. even when \$x is 0';}{    # P33 (*) Determine whether two positive integer numbers are coprime.    #     # Two numbers are coprime if their greatest common divisor equals 1.    # Example:    # * (coprime 35 64)    # T    sub coprime(Int \$a, Int \$b) { gcd(\$a,\$b) == 1}    ok coprime(35,64), "We should be able to tell that 35 and 64 are coprime";    ok coprime(64,35), ".. and viceversa";    ok !coprime(13,39), ".. but 13 and 39 are not";}{    sub totient_phi(Int \$num) {        +grep({gcd(\$_,\$num) == 1}, 1 .. \$num);    }    # TODO: s/my/constant/    my @phi = *,1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8;    # from Sloane OEIS A000010    for 1..20 -> \$n {        is @phi[\$n], totient_phi(\$n), "totient of \$n is @phi[\$n]";    }}{    # P35 (**) Determine the prime factors of a given positive integer.    #    # Construct a flat list containing the prime factors in ascending order.    # Example:    # * (prime-factors 315)    # (3 3 5 7)    sub prime_factors(\$n is copy) {        my @factors;        my \$cand = 2;        while (\$n > 1) {            if \$n % \$cand == 0 {                @factors.push(\$cand);                \$n /= \$cand;            }            else {                \$cand++;            }        }        return @factors    }    is prime_factors(315), (3,3,5,7), 'prime factors of 315 are 3,3,5,7';}{    # 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)    # ((3 2) (5 1) (7 1))    #     # Hint: The problem is similar to problem P13.        our sub prime_factors_mult(Int \$n is copy){      return () if \$n == 1;      my \$count = 0;      my \$cond = 2;      return gather {        while \$n > 1 {          if \$n % \$cond == 0 {     \$count++;     \$n div= \$cond;          }          else {     if \$count > 0 {     take [\$cond,\$count];     \$count = 0;     }     \$cond++;          }        }        take [\$cond,\$count];      }    }    is prime_factors_mult(1),(), "We ignore 1";    is prime_factors_mult(2),([2,1]), "We get prime numbers prime";    is prime_factors_mult(4),([2,2]), ".. and multiplicity right";    is prime_factors_mult(12),([2,2],[3,1]), ".. and products of primes";    is prime_factors_mult(315),([3,2],[5,1],[7,1]), ".. and ignore multiplicity 0"}{    # P37 (**) Calculate Euler's totient function phi(m) (improved).    #     # See problem P34 for the definition of Euler's totient function. If the list of    # the prime factors of a number m is known in the form of problem P36 then the    # function phi(m) can be efficiently calculated as follows: Let ((p1 m1) (p2 m2)    # (p3 m3) ...) be the list of prime factors (and their multiplicities) of a given    # number m. Then phi(m) can be calculated with the following formula:    #     # phi(m) = (p1 - 1) * p1 ** (m1 - 1) + (p2 - 1) * p2 ** (m2 - 1) + (p3 - 1) * p3 ** (m3 - 1) + ...    #     # Note that a ** b stands for the b'th power of a.        # This made me mad, the above formula is wrong    # where it says + it should be *    # based on the fact that    # phi(prime**m)=prime**(m-1)*(prime-1)    # and    # some_number=some_prime**n * some_other_prime**m * ....         sub phi(\$n) {      my \$result=1;            # XXX - I think there is a way of doing the unpacking + assignment       # in one step but don't know how          for prime_factors_mult(\$n) -> @a {        my (\$p,\$m) = @a;        \$result *= \$p ** (\$m - 1) * (\$p - 1);      }      \$result;    }            my @phi = *,1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8;    for 1..20 -> \$n {        is phi(\$n), @phi[\$n], "totient of \$n is {@phi[\$n]}";    }}{    # P38 (*) Compare the two methods of calculating Euler's totient function.    #     # Use the solutions of problems P34 and P37 to compare the algorithms. Take the    # number of logical inferences as a measure for efficiency. Try to calculate    # phi(10090) as an example.    skip 'No Benchmark module yet', 1}{    # 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.        our sub primes(\$from, \$to) {        my @p = (2);        for 3..\$to -> \$x {            push @p, \$x unless grep { \$x % \$_ == 0 }, 2..ceiling sqrt \$x;        }        grep { \$_ >= \$from }, @p;    }        is primes(2,11), (2,3,5,7,11), "a few.";    is primes(16,100), (17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97), "a few more.";}{    # 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:    # * (goldbach 28)    # (5 23)        sub goldbach(\$n) {        my @p = primes(1, \$n-1);        for @p -> \$x {            for @p -> \$y {                return (\$x,\$y) if \$x+\$y == \$n;            }        }    }        is goldbach(28), (5, 23), "Goldbach works.";}# vim: ft=perl6`
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