Chapter 1.D, Problem 68E

Twin Primes Conjecture. If you write out the first several prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, …) you will see that occasionally the gap between two consecutive primes is 2 (for example, 5, 7 and 17, 19). These pairs of closely spaced primes are called twin primes. A famous conjecture states that the number of twin primes is infinite. While a deductive proof of this conjecture has never been found, twin primes with nearly 400,000 digits have been identified, which gives experimental support to the conjecture. Find the first ten pairs of twin primes and present an inductive proof for the truth of the conjecture. Do you think the conjecture is true? Why or why not?