Arithmetic sequences of prime numbers

Today, I saw a question that goes as follows:

If \(\{a,b,c,d,e\}\) is an increasing arithmetic sequence of five prime numbers, what is the least possible value of \(|a-b|\)?

Let’s write the sequence as \(\{a, a+k, a+2k, a+3k, a+4k\}\), where \(k \in \mathbb{N}\). It is easy to see that \(a\neq 2\). It is also easy to see that \(k\) cannot be odd.

Assume that \(k=2\) and consider the remainders of \(\{a, a+2, a+4, a+6, a+8\}\) when divided by \(3\). Since \(a\) cannot be \(3\) (as \(a+6\) is a prime), \(a\) must be of the form \(3m+1\) or \(3m+2\) and this would imply that either \(a+2\) or \(a+4\) is prime. This contradiction comes from the assumption that \(k=2\).

Assume that \(k=4\) and consider the remainders of \(\{a, a+4, a+8, a+12, a+16\}\) when divided by \(5\). Since \(a\) cannot be \(5\) (as \(a+4\) is a prime), \(a\) must be of the form \(5m+1\), \(5m+2\), \(5m+3\), or \(5m+4\) and the same argument as above shows that \(k\neq 4\).

Now we know that \(k\geq 6\). Indeed the least possible value of \(k\) turns out to be \(6\) because \(\{5, 11, 17, 23, 29\}\) consists of primes.

library(primes)
max <- 2000
by = 6
length.out = 5
p_seq <- generate_primes(max = max)
for (i in seq(1, length(p_seq))) {
  candidate <- seq(from = p_seq[i], by = by, length.out = length.out)
  call <- all(candidate %in% p_seq)
  if (call) {
    print(candidate)
  }
}

## [1]  5 11 17 23 29

An argument similar to above can be used to show that one of \(\{a, a+6, a+12, a+18, a+24\}\) is divisible by \(5\), so \(\{5, 11, 17, 23, 29\}\) is the only arithmetic sequence of primes with \(5\) terms with a common difference of \(6\).