Problem:
The positive difference between a pair of primes is equal to , and the positive difference between the cubes of the two primes is . What is the sum of the digits of the least prime that is greater than those two primes?
Answer Choices:
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Solution:
Let and be the two primes. Then . Expanding and simplifying yields . Adding 1 to both sides and factoring gives . The fact that suggests trying , and indeed that works. The next prime after the two primes 71 and 73 is 79 . The requested sum of digits is .
Note: A pair of primes is called twin primes if their positive difference equals 2. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the Twin Prime Conjecture).
The problems on this page are the property of the MAA's American Mathematics Competitions