This is a game for 2 players.
Draw a single column of n squares.
Players take turns. On a turn a player writes an x into any blank square. Each player makes a total of floor(n/4) x's, so about half the squares are filled at game's end.
A player's score equals the number of DISTINCT primes they can find whose binary representation is a substring in the columns, if each x is interpreted as a 1 and each blank square is interpreted as a 0. Player 1 finds binary primes written from top to bottom in the column, and player 2 finds primes written from bottom to top in the column.
Largest score wins.