2. The Goldbach conjecture (1742) is one of the oldest and best-known unsolved problems in number theory (and all of mathematics). It states that every even integer greater than 2 can be expressed as the sum of two primes.

(a) Compose a function that determines for an input variable 2N (even number), whether it can be split as the sum of two prime numbers: 2N = p1 + p2. Use the Matlab function isprime in your tests, type help isprime for more information.

(b) (1) Check the Goldbach conjecture for all even numbers smaller than 1000. (2) Convince yourself that your method in (a) is efficient. If so, check the Goldbach conjecture for all even numbers up to 105 .

(c) For every even number smaller than 1000, determine the number of possible distinct splittings. For example, 10 = 5 + 5 = 7 + 3(= 3 + 7), hence 10 has two distinct splittings. Store the numbers of distinct splittings in an array. Which even numbers smaller than 1000 only have one splitting? Which even number has the largest number of splittings?