On three questions concerning \(0,1\)-polynomials (Q873838)

The three main results proved in this paper are the following: 1) For every integer \(n\geq36\) the polynomial \( X^n+ X^{35}+ X^{34}+ X^{33}+ X^{32}+ X^{16}+ X^{15}+ X^{3}+1 \) is irreducible over the integers. 2) If \(a<b<c\) are positive integers then at least one of the polynomials \(X^b+X^a+1\), \(X^c+ X^b+1\) and \(X^c+X^a +1\) is irreducible over the integers. 3) There exists an infinite set of positive integers \(S\) such that if \(\{e_1,\ldots, e_n \}\) is any finite subset of \(S\) then the polynomial \( X^{e_1}+X^{e_2}+\cdots+X^{e_n}+1 \) is irreducible over the integers.