Pierre Kaplan

Pell's equations X2 − mY2 = −1, −4 and continued fractions

Abstract Let m denote a positive nonsquare integer. It is shown that if Pells equation X 2 − mY 2 = −1 is solvable in integers X and Y then the equation X 2 − mY 2 = −4 is solvable in coprime integers X and Y if and only if l(√m) ≡ l( 1 2 (1+√m)) (mod 4), where l ( α ) denotes the length of the period of the continued fraction expansion of the quadratic irrational α.

An Artin character and representations of primes by binary quadratic forms II

For any squarefree positive m there exists exactly one solvable antipellian equation, which can be used to construct a certain dihedral extension L/Q, cyclic of degree 4 above k=Q(√−m). We calculate the conductor of L/k and the value of the Artin character of L/k on the corresponding congruence ideal classes of order 2 of k. From this, we deduce results for the representations of powers of primes by binary quadratic forms, in the case where the norm of the fundamental unit of Q(√m) is +1.