Spoiler for New Scientist Enigma 1740: “Sudoprime” (Follow the link to see the puzzle.)
Label the squares:
A - - - B C D - E F - G H I - J K - L M N - - - O
A is 5 and H is 9. C, D, F, I, K, L, M, N, and O must be odd and not 5.
AC must be 53 or 59. Try 59 first:
Then CD must be 97. K cannot be 7 or 9 so must be 1 or 3. Then DGK is 743, 751, or 761. L cannot be 7 or 9 so must be 1 or 3. If DGK is 743 then GHI is 491 and L cannot be 1 or 3. Dead end. If DGK is 751 then GHI is 593 and again L cannot be 1 or 3. Dead end. So DGK is 761, GHI is 691, and L is 3. EIL must be 613. This means O must be 1. F cannot be 1 or 7 or 9 so must be 3; then EF can only be 63 which is not prime. Dead end.
So AC must be 53. CD can be 31 or 37. Try 37:
Same argument as above except we conclude F must be 9, so EF is 69 which is not prime. Dead end.
Then CD is 31. N cannot be 3 or 9 so must be 1 or 7. K, L, and O must be 3 or 7. So LM and MO are 31 and 17 or 71 and 13. Either way M is 1. Try 31 and 17:
Then F is 9. I is 1 or 7. The 3-digit primes ending in 13 or 73 are 113, 173, 313, 373, 613, 673, 773. None is legal for EIL.
So LM and MO are 71 and 13. Then I is 1 or 3. The 3-digit primes ending in 17 or 37 are 137, 317, 337, 617, 937. Only 617 is legal. Then EF is 67. K is 3. GHI is 491 or 691. But 143 is not prime so DGK is 163. B is 4. The only possibilities for JN are 41 and 47, but we already have BF = 47 so JN is 41.
The grid is:
5 - - - 4 3 1 - 6 7 - 6 9 1 - 4 3 - 7 1 1 - - - 3
and the answers required are 617 and 41.