Solutions can be found here.
(**) Determine whether a given integer number is prime.
(**) Determine the prime factors of a given positive integer.
- Construct a flat list containing the prime factors
in ascending order.
?- prime_factors(315, L).
L = [3,3,5,7]
(**) Determine the prime factors of a given positive integer (2).
- Construct a list containing the prime factors and
?- prime_factors_mult(315, L).
L = [[3,2],[5,1],[7,1]]
Hint: The solution of problem 1.10 may be helpful.
(*) 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.
(**) 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.
?- goldbach(28, L).
L = [5,23]
(**) 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.
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):
992 = 73 + 919
1382 = 61 + 1321
1856 = 67 + 1789
1928 = 61 + 1867
(**) Determine the greatest common divisor of two positive integer
- Use Euclid's algorithm.
?- gcd(36, 63, G).
G = 9
Define gcd as an arithmetic function; so you can use it like this:
?- G is gcd(36,63).
G = 9
(*) Determine whether two positive integer numbers are coprime.
- Two numbers are coprime if their greatest common divisor equals 1.
?- coprime(35, 64).
(**) Calculate Euler's totient function phi(m).
- Euler's so-called totient function phi(m) is defined as the number
of positive integers r (1 <= r < m) that are coprime to m.
Example: m = 10: r = 1,3,7,9; thus phi(m) = 4.
Note the special case: phi(1) = 1.
?- Phi is totient_phi(10).
Phi = 4
Find out what the value of phi(m) is if m is a prime number.
Euler's totient function plays an important role in one of the
most widely used public key cryptography methods (RSA). In this
exercise you should use the most primitive method to calculate
this function. There is a smarter way that we shall use in 2.10.
(**) Calculate Euler's totient function phi(m) (2).
- See problem 2.09 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 2.03 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.
(*) Compare the two methods of calculating Euler's totient function.
- Use the solutions of problems 2.09 and 2.10 to compare the algorithms.
Take the number of logical inferences as a measure for efficiency.
Try to calculate phi(10090) as an example.