Turn based Number game

Question

You and a friend take turns saying positive integers greater than one. You can't say a number that is a multiple of previous numbers or a sum of multiples of previous numbers. For example, if A says 3 and B says 7 then A cannot say 17, because 17=2*7+3. If you are unable to say a number you lose.

One way to lose is by starting off saying the number 2 or 3; then the other person says the one you didn't say and when it comes back to your turn you lose (2 and 3 can make any positive integer greater than 1).

What are some strategies, how do you win?

Answer 1

I am pretty sure this is the

Sylver coinage game. The example winning strategy there is to say 5, 7, 11, 13 etc. as a first move (diagram coming soon for 5) Diagram cancelled - see this

"Unlike many similar mathematical games, sylver coinage has not been completely solved, mainly because many positions have infinitely many possible moves. Furthermore, the main theorem that identifies a class of winning positions, due to R. L. Hutchings, guarantees that such a position has a winning strategy but does not identify the strategy. Hutchings's Theorem states that any of the prime numbers 5, 7, 11, 13, …, wins as a first move, but very little is known about the subsequent winning moves: these are the only winning openings known."

-- Wikipedia