Wieferich Primes and Monogenic Trinomials

A prime p p p is called a Wieferich prime if 2 p − 1 ≡ 1 ( mod p 2 ) 2ᵖ⁻¹≡ 1 p² 2 Superscript p minus 1 Baseline identical to 1 left parenthesis mod p squared right parenthesis . A monic polynomial f ( x ) ∈ Z [ x ] f(x)∈ Z\, [x] f left parenthesis x right parenthesis element of double struck upper Z left bracket x right bracket of degree N ≥ 2 N≥ 2 upper N greater than or equals 2 is called monogenic if f ( x ) $f(x)$ f left parenthesis x right parenthesis is irreducible over Q Q double struck upper Q and { 1 , θ , θ 2 , … , θ N <