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'Prime Sequences' printed from http://nrich.maths.org/
In 2004 an exciting new result was
proved in Number Theory by two young mathematicians Ben Green and
Terrence Tao. They proved that if you look in a long enough list of
the prime numbers then you will be able to find numbers which form
an arithmetic progression containing as many numbers as you choose!
In this question we explore some of the interesting issues
surrounding arithmetic progressions of prime numbers.
An $AP-k$ sequence is $k\geq 3$ primes
in arithmetic progression. See examples
This problem involves several linked parts leading up to a final
challenge. Try some of the earlier questions to gain insights into
the final challenge. These can be attempted in any order. You might
find that you naturally ask yourself questions which are found
later in the list of questions and you might find that one part
helps in the consideration of another part. Of course, you are
welcome to go straight to the final challenge. However, you might
also wish to start with one of the earlier challenges and see how
many of the other challenges you naturally discover whilst
exploring the underlying mathematical structure.
When you have thought about some of the previous problems you might
like to try the final challenge
In doing these problems you might like to see this list