Problem Statement

When calculating prime numbers > 2, the difference between adjacent primes is always an even number. This difference, also referred to as the gap, varies in an random pattern; at least, no pattern has ever been discovered, and it is strongly conjectured that no pattern exists. However, it is also conjectured that between some two adjacent primes will be a gap corresponding to every positive even integer.

This task involves locating the minimal primes corresponding to those gaps. Though every gap value exists, they don't seem to come in any particular order. For example, this table shows the gaps and minimum starting value primes for 2 through 30:

For the purposes of this task, considering only primes greater than 2, consider prime gaps that differ by exactly two to be adjacent.

For each order of magnitude m from 10¹ through 10⁶:

For an m of 10¹; The start value of gap 2 is 3, the start value of gap 4 is 7, the difference is 7 - 3 or 4. 4 < 10¹ so keep going. The start value of gap 4 is 7, the start value of gap 6 is 23, the difference is 23 - 7, or 16. 16 > 10¹ so this the earliest adjacent gap difference > 10¹.

Note: the earliest value found for each order of magnitude may not be unique, in fact, is not unique; also, with the gaps in ascending order, the minimal starting values are not strictly ascending.

Let's start with the solution:

lowpgap=: {{
  magnitude=. ref=. 10^y
  whilst. -.1 e. ok do.
    magnitude=. 10*magnitude
    g=. 2-~/\ p=.p: i.magnitude
    mag=. p{~}.g i. i.&.-: >./g NB. minimum adjacent gaps
    dif=. 2-~/\ mag
    ok=. ref < dif
  end.
  (0 1+ok i.1){mag
}}


   lowpgap 1
7 23
   lowpgap 2
113 1831
   lowpgap 3
113 1831
   lowpgap 4
9551 30593
   lowpgap 5
31397 404597
   lowpgap 6
396733 1444309
   lowpgap 7
2010733 13626257