Can You Solve This Math Problem from the Asian International Mathematical Olympiad (AIMO)?
It’s Actually Simpler Than You Think
Problem
What is the maximum number of terms in an arithmetic sequence of primes with a common difference of 6?
Pause for a moment and try to solve this problem yourself! 🧠 🧠 ✏️
Solution
Got your answer? Ok! Here’s the solution! 3…2….1..
We know that the prime numbers p, p + 6, p + 12,…, form an arithmetic sequence with a common difference. 6.
After dividing the terms by 5, we write the remainders.
There are five possible sequences of remainders: 0, 1, 2, 3, 4, 0,…; 1, 2, 3, 4, 0, 1,…;2, 3, 4, 0, 1, 2,…; 3, 4, 0, 1, 2, 3,…; and 4, 0, 1, 2, 3, 4...
Unless it is 5, the term that corresponds to remainder 0 will not be prime. Thus, starting with 5 is the only method to obtain an arithmetic sequence of primes with a common difference of 6.As a result, the order is 5, 11, 17, 23, 29, 35,…
There are only five prime phrases. Thus, the most terms that can be used are five.
Final Answer
The maximum number of terms in an arithmetic sequence with a common difference of 6 is 5.
