N Certain Rational Expressions whose Prime Divisors are Cubic Residues (Mod P)

Abstract

Suppose p is a positive prime number congruent to 1 modulo 3. Then, as is well known, one can write $4p = {L^2} + 27{M^2} = \lambda \bar \lambda ,{\text{ where }}\lambda = L + 3M\sqrt { - 3} ,L,M \in Z,L \equiv 1\left( {\bmod 3} \right),L \equiv M$ (mod 2). The integers L and M are uniquely determined by these conditions. For n ≥ 1, write 4p ⁿ = f ⁿ(L, M)² + 27 g ⁿ(L,M)², f ⁿ (L, M) and g ⁿ (L,M) being rational integers defined by

$$ {f_n}\left( {L,M} \right) = \frac{{{{\left( {L + 3M\sqrt { - 3} } \right)}^n} + {{\left( {L - 3M\sqrt { - 3} } \right)}^n}}}{{{2^n}}},\,{g_n}\left( {L,M} \right) = \frac{{{{\left( {L + 3M\sqrt { - 3} } \right)}^n} - {{\left( {L - 3M\sqrt { - 3} } \right)}^n}}}{{3\sqrt { - 3} {2^n}}} $$