Bi-twin prime chains

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I mentioned bi-twin prime chains in the previous post, but didn’t say much about them so as not to interrupt the flow of the article.

A pair of prime numbers are called twins if they differ by 2. For example, 17 and 19 are twin primes.

A bi-twin chain is a sequence of twin primes in which the average of each twin pair is double that of the preceding pair. For example,

5, 7, 11, 13

is a bi-twin chain because (5, 7) and (11, 13) are twin pairs of primes, and the average of the latter pair is twice the average of the former pair.

There is some variation in how to describe how long a bi-twin chain is. We will define the length of a bi-twin chain to be the number prime pairs it contains, and so the chain above has length 2. The BiTwin Record page describes a bi-twin chain of length k as a chain with k − 1 links.

It is widely believed that there are infinitely many twin primes, but this has not been proven. So we don’t know for certain whether there are infinitely many bi-twin prime chains of length 1, much less chains of longer length.

The bi-twin chain of length three with smallest beginning number is

211049, 211051, 422099, 422101, 844199, 844201

Bi-twin records are expressed in terms of primorial numbers. I first mentioned primorial numbers a few days ago and now they come up again. Just as n factorial is the product of the positive integers up to n, n primorial is the product of the primes up to n. The primorial of n is written n#. [1]

The longest known bi-twin chains start with

112511682470782472978099, 112511682470782472978101

and has length 9.

We can verify the chains mentioned above with the following Python code.

def bitwin_length(average):
    n = average
    c = 0
    while isprime(n-1) and isprime(n+1):
        c += 1
        n *= 2
    return c

for n in [6, 211050, 112511682470782472978100]:
    print(bitwin_length(n))

[1] There are two conventions for defining n#. The definition used here, and on the BiTwin Record page, is the product of primes up to n. Another convention is to define n# to be the product of the first n primes.

The priomorial function in Sympy takes two arguments and will compute either definition, depending on whether the second argument is True or False. The default second argument is True, in which case primorial(n) returns the product of the first n primes.

The post Bi-twin prime chains first appeared on John D. Cook.

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