Problem of the Month

### Twin Primes

Factors are numbers that divide other numbers without remainder. Five (5) is a factor of 35 because 35/5 = 7 with no remainder. Prime numbers have only two factors: they are one (1), which divides all numbers, and the number itself. For example, 19 is prime and has factors of 1 and 19 only.

1. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. The pairs like 5 and 7, and 29 and 31, are called twin primes because they differ by only two. What is the next pair of twin primes greater than 31?

2. Inspect the set of numbers that fall between several pairs of twin primes. Do they have common factors? Can you prove that the numbers between every pair of twin primes greater than 3 will be a multiple of 6?

HINT: The numbers from one multiple of six to the next can be represented by 6n, 6n + 1, 6n + 2, 6n + 3, 6n + 4, 6n + 5, and 6n + 6. Which of these formulas could represent primes?

solutions

Read about the search for the proof that twins are infinite in number.