It is possible for any value equal to or greater than 3, assuming that you have the ability to chop the coins into pieces with infinite precision. You do not need to have coins which have an even weight distribution.
For even numbers, you divide into two groups and weigh. Set aside the lighter half as L, the heavier side as H. Now split H into two parts and weigh. If they are equal, then H will disappear and you know that the fake is lighter. If H1 and H2 are unequal, then L will disappear and you know that the fake is heavier. Continue splitting and weighing until you have two coins or one coin left.
If your second weighing showed that the fake was heavier, then the fake is the heavier of the remaining coin(s). If your second weighing showed that the fake was lighter, then the fake is the lighter of the remaining coin(s).
For odd numbers, it is possible, using deduction and limits.
- Coins can be split into pieces without losing material.
- Coins can be weighed relative to themselves without proving anything. If I split a coin into pieces, I can continue adjusting the split until I have evenly divided the coin, and none of this causes the coin to disappear.
So, take all the coins and then weight 2v1 until you have weighed every single combination of coins. If at any point 1 is heavier than 2, then all other coins will disappear and you know which coin is fake and that it's heavier.
Okay, now what if you don't find the fake? Split every coin into two equal parts (by weight), keeping track of which coins parts belong to which coin. Now weigh two halves of one coin against three halves of other coins until you have tested all combinations. If at any point a coin is heavier than its opposition, then the fake will be revealed and known to be heavier.
So what we have now is a formula of N-1/N, where N is the number of pieces we've split each coin into. Take the limit as N approaches infinity and we have 1, which represents equality of weight, which is not possible. So, for any given threshold of measurement, we can either find the fake and know it's heavier or find nothing and prove that the fake is lighter.
The key here is that any standard of accuracy is reachable, if we have a great deal of patience. In fact, if we know the standard of accuracy in advance, we can simply chop the coins into sufficiently-small pieces and do one round of weighings.
Knowing that fake is lighter (if it were heavier we would have found it), we can solve for an odd number of coins. Weigh the coins 1v1 until you have a single coin left standing. That coin is the fake, and you know it's lighter. For higher values, you can be more efficient by weighing larger piles against each other, but the end result is that you can narrow it down to 1 coin, and then you win.