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 ....

40 Fanayiotis G. Tsangaris (n = 1, 2,...) where + mVd is the fundamental solution of (1.1) and X\ + ywfd is the fundamental solution of (P) x

2 - dy

2 = 1 (d^ •). The following Theorems can be found in [5] (cf. also [4]). Theorem 1.2. Consider the Diophantine equation (F) X

2 - dY

2 =C. (d^D, C > 0). Let X*+Y*y/d be the fundamental solution of a class A

r of integral solutions of (F) with X* > 0 Let x

n + y

n\fd

, where n = 0,1, . . be the sequence of all non-negative integral solutions of (P). Let X

n + Y

nVd = [x; + Y

r*Vd){x

n + y

nVd) for all n = 0,1,..., x'

n -f Y^Vd = (x; - Y;Vd){x

n + y

ny/d) for all n = 1,2,... (for a typical r

). Then the following hold true: (i) Y

n

+1 > Y

n > 0 for every n - 0,1,.... (ii) Let Y; > 0. Then >Y

n>Y

n> 0 for every n = 1, 2,.... (Hi) Let Y; = 0. Then = Y'

n for every n = 0,1,.... fivj Let be genuine (= non-ambiguous). Then >Y

n>Y^> 0 for all n = 1,2,.... (V) Let A

r be ambiguous. Then for every m there exist n such that: X'm = Xn and ^m = ^n • (Vi J Let Ä

r* + Y*y/d, where r = 1, 2,..., m, be the only integral solutions of (F) such that 0 < x; < y/(xi + l)C/2 and 0< Y

r* < y

x VC/+ 1). Then the set of all non-negative integral solutions of (F) consists of all pairs (X

n

, Y

n) together with all pairs (X

n,Y

n

) for all respective genuine classes A

r in addition to all pairs (X

n,Y

n) for all respective ambiguous classes