The sequence of natural numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,… does not need any introduction. These are prime numbers. A beautiful 2300 year old result by Euclid tells us that there are infinitely many primes. In other words the above sequence never terminates. The family of prime numbers contains many pairs of twins and cousins! If a number p and p+2 both are primes then the pair (p, p+2) is called a pair of twins. For example (3, 5), (5, 7), (11, 13),….., (659, 661),…. are pairs of twins. On the similar lines if p and p+4 both happen to be prime then the pair (p, p+4) is called a pair of cousins and if p and p+6 are both primes then the pair (p, p+6) is called a pair of sexy primes. For example (19, 23) is a pair of cousins and (31, 37) is a pair of sexy primes.
Are there infinitely many pairs of twin primes? No one till date has been able to find an answer to this seemingly easy question! The same is true with pairs of cousin primes and sexy primes.
Now a related question Find a number p such that p, p+2, p+4 all three are primes. One answer to this question is: p = 3; since 3, 5, 7 are primes. If you sit for a few minutes with paper and pencil, you will realize that p = 3 is the only answer to the above question. In other words 3, 5, 7 is the only arithmetic progression in prime numbers with a common difference 2. The length, i.e. the number of terms in this arithmetic progression is three. An arithmetic progression of primes with length four is 5, 23, 41, 59; the common difference being 18.
How long can the arithmetic progressions of prime be? This problem bothered mathematicians foryears until 2004 when Ben Green and Terence Tao came up with a surprisingly elegant answer “as you please!” That means you take any number, say as “small” as 10 million and one should be able to find an arithmetic progression of length 10 million with each term a prime number! The result is now popular by the name GreenTao Theorem. Interestingly the methods of Tao and Green do not give us any recipe to cook arbitrarily long arithmetic progressions in primes, still they do confirm the existence of such progressions. That’s why the GreenTao Theorem is an example of what are called existential theorems in Mathematics.
The longest known arithmetic progression of primes has length 25. Raanan Chermoni and Jaroslaw Wroblewski were enthusiastic enough to find the following arithmetic progression of primes: 6171054912832631 + 366384 × 223092870 × n (where n varies from 0 to 24) !!! They worked it out recently in May 2008. How did they do it? Well, they were running clever computer programs for hundreds of CPU years!The hunt for an arithmetic progression in primes with 26 terms is on.
It is amazing that we start recognizing natural numbers as soon as we learn to walk, we grow up with numbers, we use them more often than anything else and yet so many mysteries in our number system lay hidden and are waiting for someone to uncover them.