How to distinguish all colors?

From the fact that the trichromatic system can distinguish more colors than the dichromatic system, which in turn can distinguish more colors than the monochromatic system, it seems obvious that the more different types of photoreceptors the vision system has, the more values are available to represent a color, and the more comparisons can be made to distinguish more colors. As trichromatic system still cannot tell all possible colors apart, how many different types of receptors are necessary to do so?

The key here is to be able to uniquely represent in the visual system the energy spectral distribution $L(\lambda)$ of an incoming light, which determines the physical property of the light (but not necessarily its perceived color). To approach the problem, we discretize the continuous spectral distribution $L(\lambda)$ by approximating it by $m$ values $L(\lambda_j),\;\;(j=1, \cdots, m)$, each representing the energy contained in one of $m$ segments of the distribution. Now the response of the $ith$ receptor to the light becomes

$$r_i=\int S_i(\lambda) L(\lambda) d\lambda \approx \sum_{j=1}^m S_i(\lambda_j)L(\lambda_j) \;\;\;\;(i=1, \cdots, n)$$

where $m$ is the number of segments used to approximate the spectral distribution and $n$ is the number of different types of receptors. To obtain the spectral distribution $L(\lambda_j),\;\;(j=1, \cdots, m)$ from the receptor responses $r_i,\;\;(i=1, \cdots, n)$ available in the visual system, it is mathematically obvious that $n$ has to be equal to $m$ so that we have $n=m$ equations which can be solved to uniquely determine the $m$ unknowns $\L (\lambda_j)$.

The number $m$ of discrete values $L(\lambda_j)$ depends on the details of the spectral distribution of the incoming lights, and usually $m$ needs to be much greater than $3$ to describe the various distributions of the incoming lights. This is why $n=3 \ll m$ types of receptors are not enough to tell all colors apart.