### Programming Problem Set: 99 Problems Chapter 2

December 9, 2015 | Labs | No Comments

Ninety-nine Problems is generalized version to famous P-99: Ninety-Nine Prolog Problems collection used for teaching programming. The problems initially set for prolog but later many solutions come from various programming language. The purpose of this problem is to give us opportunity to practice our skills in logic programming. The goal is to find the most elegant solution of the given problem. Efficiency is important, but logical clarity is even more crucial.

The problem set are divided into seven categories / chapters: Lists, Arithmetic, Logic and Codes, Binary Trees, Multiway Trees, Graphs, and Miscellaneous.

In this chapter you will be only given a problem set. The solution might come however it would be on different page.

This chapter will cover about **Arithmetic**. A list is either empty or it is composed of a first element (head) and a tail, which is a list itself. As a continuation from previous chapter, the problem will be started from last previous number.

#### 29. Determine whether a given integer number is prime.

Example: is_prime_p( 7 ) -> Yes

#### 30. Determine the prime factors of a given positive integer.

Construct a list containing the prime factors in ascending order Example: prime_factor_p( 315 ) -> [ 3, 3, 5, 7 ]

#### 31. Determine the prime factors of a given positive integer (2)

Construct a list containing the prime factors and their multiplicity. Example: prime_factor2_p( 315 ) -> [ [3,2], [5,1], [7,1] ] Hint: The solution of problem 10 may be helpful.

#### 32. A list of prime number

Given a range of integers by its lower and upper limit, construct a list of all prime numbers in that range. Example: prime_list_p( 3, 15 ) -> [3, 5, 7, 11, 13 ]

#### 33. 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. Find the two prime numbers that sum up to a given even integer Example: goldbach_p( 28 ) -> [ 5, 23]

#### 34. 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_p( 9, 20 ) 10 = 3 + 7 12 = 5 + 7 14 = 3 + 11 16 = 3 + 13 18 = 5 + 13 20 = 3 + 17 In most case, 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.

#### 35. Determine the greatest common divisor of two positive integer number

Use Euclid's algorithm Example: gcd_p( 36, 63 ) -> 9

#### 36. Determine whether two positive integer numbers are coprime

Two numbers are coprime if their greates common divisor equals 1 Example: coprime_p( 35, 64 ) -> Yes

#### 37. Calculate Euler’s totient function phi(m)

Euler's so-called totient phi(m) is defined as the number of pisitive integers r (1 <= r < m) that are coprime to m. If m = 10 then r = 1, 3, 7, 9; thus phi(m) = 4. Note the special case phi(1) = 1 Example: phi_p( 10 ) -> 4

#### 38. Calculate Euler’s totient function phi(m) (2)

See the previous problem for definition of Euler's totient function. If the list of the prime factors of a number m is known in the form of problem 32 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 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.

#### 39. Compare the two methods of calculating Euler’s totient function.

Use the solution of problem 37 and 38 to compare algorithm. Take the number of logical inferences as a measure for efficiency. Try to calculate phi(10090) as an example

## Solution:

- Haskell
- Lisp
- Prolog
- Python