The Diophantine equation x2 + 11 = 3n and a related sequence

AbstractIt is proven that the Diophantine equation x2 + 11 = 3n has as its only solution (x, n) = (4, 3). This establishes that the Diophantine equation x2 + D = pn, where D ≡ 3(mod 4) and p = (D + 1)4, is prime, has no nontrivial solutions (x, n) for D ≥ 11. The only remaining case, D = 7, is known to have exactly 5 solutions. It is shown that the sequence of an satisfying an+2 = an+1 − 3an (a1 = a2 = 1) has the property that no integer greater than 1 can occur more than once. This is applied to the equation x2 + 11z2 = 3n