Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

TSANGARIS, PANAYIOTIS G., A sieve for all primes of the form R2+(X-)-l)2 ....

A sieve for all primes of the form r

2 + (a-+l)

2 43 number (cf. Theorem 3.5). The set of all non-negative solutions of (Fk) is determined recursively by Theorem 3.6 together with Corollary 3.7. Proposition 3.1. Consider the Diophantine equation (Fk) where k is a natural number. Let X* + Y*V2 be a solution of(F

k). Then X* + Y~y/2 is the fundamental solution of a class of integral solutions of (Fk) if and only if the following (equivalentJ inequaUties are satisfied: Proof. By using Theorem 109 in [2]. Note. The fundamental solution of (F

0) is X*+ Y*V

2 = 1 + \/2. Proposition 3.2. Let k be a natural number. Then 2k — 1 + (k — l)\/2 is the fundamental solution of a class of integral solutions of (Fk). Proof. Evident by Proposition 3.1. Proposition 3.3. Let A be a class of integral solutions of the Diophantine equation (F), C ^ 0. Let X + Y Vd be a representative of A and Then the following hold true: (i) A is a genuine if and only if at least one of the numbers L, M is not integral. (ii) A is ambiguous if and only if both numbers L and M are integral. Proof. Immediate by using Nagell's criterion (p. 205, [2]). Theorem 3.4. Let X* + Y*y/

2 be the fundamental solution of a class A of integral solutions of (Fk), where k — 1

,2,.... Then the following hold true: (i

) A is genuine if and only ifY* > 0. (ii) A is ambiguous if and only ifY* = 0. Proof, (i) (a) If A is genuine, then the previous Proposition 3.3 easily implies Y* > 0. (b) Let now Y* > 0 and assume that A is ambiguous. Then, by the same Proposition, the numbers (3.1) (3.2) 0 < I A"" I < 2k - 1, 0 < Y* < k - 1. L = (-X

2 - dY

2)/C and M = -2XY/C. L = (-X*

2 - 2Y*

2)/(2k

2 - 1) and M = -2X*Y*/(2k

2 - 1)