I was looking at my Algebra book and there were some questions I couldn't really know how to solve. They are of similar type. Here they go:
$1.$ Prove that equation $x^{4}+3y^{4} = 777...777$ (number of the digits is $n$) has no integer solution for any positive integer $n$.
$2.$ Does the equation $x^{5}-x+y^{2} = 12345678$ have any integer solutions?
$3.$ Is there a positive integer $x$ such that $2^{x}+x^{2}=555555555$?
$4.$ Is there an integer $x$ such that $x^{2}+x=987654$?
$5.$ Is there an integer $x$ such that $x(x+1)(x+2)=98765432$?
Is there any same method for solving those problems, or is the method individual for each problem, and how to solve them? The only restriction is no modulo. Thank you in advance.