How to crack a Linear Congruential Generator (LCG) The full question is: How to crack a Linear Congruential Generator when a, c and m in the LCG formula. I googled this question and I found what I wanted:
here is the question: https://brilliant.org/problems/breaking-linear-congruential-generators/
and here is the screenshot of the answer (you need to make an account, to see the answer):
https://imgur.com/a/RrxEU
The problem is that I don't understand the answer. 
1.) I do not understand this equation from the answer (what is this equation and why is it used): $$Z_n = Y_nY_{n+2}- Y^2_{n+1}$$
2) And I don't understand how a and c are found afterwards.
Any help is apprecciated
Also I am aware that there are other methods to solve this question for example by using a matrix: Calculation for linear congruential generator: Setting up equations
Thank you
 A: Just use the formulas in the picture. 
Part 1: With $Y_n \equiv a Y_{n-1} \pmod m$ you get
$$Y_{n+1} \equiv a Y_{n} \equiv a^2Y_{n-1} \pmod m$$
$$Y_{n+2} \equiv a Y_{n+1} \equiv  a^3Y_{n-1} \pmod m$$
and therefore 
$$Z_n \equiv Y_{n}Y_{n+2} - Y_{n+1}^2 \equiv  (a\cdot a^3 - (a^2)^2) Y_{n-1} \equiv  0 \pmod m$$
Thus $Z_n = m\times z_n$ is a multiple of $m$ and (as claimed in the image) for a sufficient number of samples $\{X_n\}$ the gcd of corresponding $Z_n$ is expected to be $m$.

Part 2: If you accept $m := 1000000007$ from the gcd calculation the two linear 
$$a X_1 + b \equiv  X_2 \pmod m$$
$$a X_2 + b \equiv  X_3 \pmod m$$
congruences can be solved as follows: First compute the modular multiplicative inverse $(X_1-X_2)^{-1} \equiv 560056067 \pmod m$ (this is usually done with the
Extended Euclidean Algorithm or Euler's theorem, see Wikipedia,
for single values you can use an online calculator, e.g. https://planetcalc.com/3311/
with the input $X_1-X_2 = 587411898,  m=1000000007$ or with Wolfram Alpha
solve (720555190-133143292)x = 1 mod 1000000007)
and then 
$$a \equiv (X_2-X_3) (X_1-X_2)^{-1} \equiv 782674123\times 560056067 \equiv 1664525 \pmod m$$
$$b\equiv X_2 - a X_1 = 133143292 - 1664525\times 720555190 \equiv  13904216 \pmod m$$
This is done like the solving of a usual system of linear equations. Subtract the second from the first equation and get $a$
$$aX_1 - a X_2 \equiv a(X_1-X_2)\equiv X_2-X_3 \pmod m$$
$$a \equiv (X_2-X_3)(X_1-X_2)^{-1} \pmod m$$
and $b$ from the first equation.
