Prime stepping stones

Question

Start by placing number $1$ anywhere on an infinite square grid. Now place numbers $2, 3, 4, \ldots, K$ in order. A number $k$ can be placed if the following rules hold:

• It must be adjacent (horizontally or vertically) to the previous number $k-1$.
• It must have at least one neighbour (horizontally or vertically) number $m$ already placed such that $k+m$ is prime. Note that $m$ can be $k-1$.

What is the largest number $K$ that you can place? You can use a computer if you want.

Answer 1

Found with the assistance of a computer program. It couldn't find anything larger so I think this is an upper bound.

47

                   1  6  7
         31 30 29  2  5  8  9
47 42 41 32 27 28  3  4 11 10
46 43 40 33 26 15 14 13 12
45 44 39 34 25 16 17 18
      38 35 24 23 20 19
      37 36    22 21
 

Answer 2

I can do

43
which is about the limit of what I'm willing to do manually ;-)

       37 36 31 30 27 26 25 24
 42 41 38 35 32 29 28 15 16 23 22
 43 40 39 34 33 12 13 14 17 20 21
             10 11  4  3 18 19
              9  8  5  2  1
                 7  6