Question

Determine all primes that could be represented by the principal form with discriminant −51

Answer 1

Determine all primes that could be represented by the principal form with discriminant −51

Any:- Step :- from condition is Theorem 2, (6) = 1, Apply Theosem ui there are Xiy,KEZ With q1w,y) =kp and \_KZ = 4.123 Step :- 2 (Elimination) We hill show that the cases k=2 and K=3 Cannot Occuor . a andy are Then Suppose qcx,y) = 2p. both Even, soala,y) = 2p=o (modu) dicking the fach that e is odd. Contra- (mod 3), So Suppose qeray) = 3p. Then q lay) = 51²=0 315 Hence, wlacy)=322 =3P so (P/3)=1. As 3/0, from Condition Cail of theoriem 2, (Plz)=(5/3) =-1: Contradiction (mod a),

cannot hope to Eliminate step :3 (Reduction). Note that we the Possibility to Show that of k=u: inded, the hant are nay, tz such there that wcy) =P, and the Necessarily a (2x, 2y) = up. We must instead argue that a Representation of up by a implies a Representation of P by q. In this case, this Easy : suppose qcx,y) = up Then as and ។ are both Even, q 9/2) =P is above so In lemmas land 2 2 We collect a number Apply of congruence Restrictions that Assuming a Represents Kp. In particular 2779 forms, from FON Our he use Lemmas I in the Elimination Stee and Lemmag 2 in the Reduction Step.