Prime plus six

Number Theory Level 3

What is the largest positive integer $n$ such that you can find a prime number $x > 6$ where the following numbers are also prime?

$x+6$
$x+ 2 \cdot 6$
...
$x+ n \cdot 6$

In other words, what is the most number of times you will ever be able to add 6 to a prime number > 6 and get a prime number?

6 5 3 1 2 There is no limit on

n

0 4

1 solution

Geoff Pilling
Jan 22, 2018

Consider these numbers:

$x$
$x+6 \text{ }(n=1)$
$x+12 \text{ }(n=2)$
$x+18 \text{ }(n=3)$
$x+24 \text{ }(n=4)$

These all have a different value (modulo 5), so exactly one divides by five, and therefore is not prime. (Unless it is 5, but 5 isn't allowed since the first number must be > 6).

This gives us an upper bound of $n = \boxed{3}$ .

Also, we can achieve this with the numbers 11, 17, 23, and 29.

Nice solution. My approach was a slight variation - I looked at the possible unit digits for primes greater than 6, namely 1, 3, 7 and 9 and looked at the subsequent unit digits after adding 6 successively: 1, 7, 3, 9, 5, 1, 7, 3, 9, .... . Since a prime (other than 5) can never end in 5 the longest possible sequence starting at a prime greater than 6 will occur if we start at a prime ending in 1, and indeed (11, 17, 23, 29) serves as an example. If we had been allowed to start at 5 then we could have bumped n up to 4. :)

The next possible example is (41, 47, 53, 59), followed by (61, 67, 73, 79), (251, 257, 263, 269), (601, 607, 613, 619) and then (1091, 1097, 1103, 1109). I suspect that there may be an infinite number of examples but I'm not sure how to prove it.

Brian Charlesworth - 3 years, 4 months ago

Ah, interesting... I wonder if there might be an elegant proof of its "infiniteness"...

Geoff Pilling - 3 years, 4 months ago

Prime plus six

1 solution

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