Can two different distributions have the same value of mean, variance, skewness, and kurtosis? Assuming that you have two discrete population distributions. 
Can they have identical values of mean ,variance, skewness and kurtosis while being different in shape visually ?
Do these four values act like a fingerprint of any distribution?
 A: Xi'an's answer proved (or at least hinted a proof) that there are different distributions with the same mean, variance, skewness and kurtosis. I just want to show an example of three visually distinct discrete distributions with the same moments (mean=skewness=0, variance=1 and kurtosis=2):

The code to generate them is:
library(moments)

n <- 1e6

x <- c(-sqrt(2), 0, +sqrt(2))
p <- c(1,2,1)
mostra1 <- sample(x, size=n, prob=p, replace=TRUE)

x <- c(-1.4629338416371, -0.350630832572269, 0.350630832573386, 1.46293384163564)
p <- c(1, 1.3, 1.3, 1)
mostra2 <- sample(x, size=n, prob=p, replace=TRUE)

x <- c(-1.5049621442915, -0.457635862316285, 0.457635862316022, 1.50496214429192)
p <- c(1, 1.6, 1.6, 1)
mostra3 <- sample(x, size=n, prob=p, replace=TRUE)

mostra <- rbind(data.frame(x=mostra1, grup="a"),
                data.frame(x=mostra2, grup="b"),
                data.frame(x=mostra3, grup="c"))
aggregate(x~grup, data=mostra, mean)
aggregate(x~grup, data=mostra, var)
aggregate(x~grup, data=mostra, skewness)
aggregate(x~grup, data=mostra, kurtosis)

library(ggplot2)
ggplot(mostra)+
  geom_histogram(aes(x, fill=grup), bins=100)

A: Take a mixture of two Normal distributions with density
$$f(x|\mu_1,\mu_2,\sigma_1,\sigma_2,\omega)=
\frac{\omega}{\sqrt{2\pi}\sigma_1}\exp\{-(x-\mu_1)^2/2\sigma_1^2\}+
\frac{1-\omega}{\sqrt{2\pi}\sigma_2}\exp\{-(x-\mu_2)^2/2\sigma_2^2\}$$
This distribution has five parameters constrained by four equations
\begin{align*}
\mathbb{E}[X]&=\omega\mu_1+(1-\omega)\mu_2\\
\text{var}(X)&=\omega\sigma_1^2+(1-\omega)\sigma_2^2+\omega(\mu_1-\mathbb{E}[X])^2+(1-\omega)(\mu_2-\mathbb{E}[X])^2\\
\mathbb{E}[X^3]&=\ldots\\
\mathbb{E}[X^4]&=\ldots
\end{align*}
 Assuming these equations are compatible, there is therefore an infinite number of solutions $(\mu_1,\mu_2,\sigma_1,\sigma_2,\omega)$.
