What is the maximum length of a sequence of primes separated by 6?

5,11, 17, 23 and 29 is a sequence of 5 primes. In a sequence of five integers separated by six, one of the integers has to be divisible by 5. 5 is the only prime that is divisble by 5. Q.E.D.

This is just an elaboration on the solution, provide by Randolph Herber. Take the longest such sequence (with length $n$ ) to be $\{p+6.k\}_{k=0}^{n-1}$ , with $p$ being the smallest prime in the sequence. Calculate all the member of the sequence modulo $5$ . They would be of form $\{p+k\}_{k=0}^{n-1}$ . The first five element have different values, modulo $5$ , so one them should at least be $0$ , modulo $5$ , which means it should be divisible by $5$ . So, the sequence can have a maximum of $4$ elements, unless it starts with $5$ , that is the only prime multiple of $5$ . We check the sequence of length $5$ , starting with $5$ na dit actually consists of all primes.