If $$\alpha$$, $$\beta$$ are roots of the equation $${x^2} + 5(\sqrt 2 )x + 10 = 0$$, $$\alpha$$ > $$\beta$$ and $${P_n} = {\alpha ^n} - {\beta ^n}$$ for each positive integer n, then the value of $$\left( {{{{P_{17}}{P_{20}} + 5\sqrt 2 {P_{17}}{P_{19}}} \over {{P_{18}}{P_{19}} + 5\sqrt 2 P_{18}^2}}} \right)$$ is equal to _________.

Let $$\alpha$$ and $$\beta$$ be two real numbers such that $$\alpha$$ + $$\beta$$ = 1 and $$\alpha$$$$\beta$$ = $$-$$1. Let p_n = ($$\alpha$$)ⁿ + ($$\beta$$)ⁿ, p_n$$-$$1 = 11 and p_n+1 = 29 for some integer n $$ \ge $$ 1. Then, the value of p$$_n^2$$ is ___________.

The sum of 162^th power of the roots of the equation x³ $$-$$ 2x² + 2x $$-$$ 1 = 0 is ________.

The number of the real roots of the equation $${(x + 1)^2} + |x - 5| = {{27} \over 4}$$ is ________.