Problem:
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
Solution:
The common difference of such a sequence must be even, for otherwise at least one of the first five terms would be even and greater than .The common difference must also be divisible by ,for otherwise at least one of the first five terms would be divisible by and greater than .Therefore the common difference must be divisible by ,and the first term of the sequence must be relatively prime to .Because the sequence consists exclusively of primes, it follows that the desired prime is .
The problems on this page are the property of the MAA's American Mathematics Competitions