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 ....

44 Fanayiotis G. Tsangaris are integers. In particular, because L is an integer it follows that (2k

2 - 1) I X*

2 + 2V

7*

2 = 4Y*

2 + 2k

2 - 1. Thus (2k

2 - 1) I 4y*

2. Also, Y* < y/(2k

2 - l)/2, i.e. 4y"

2 < 2(2k

2 - 1). Hence 2k

2 — 1 < 4F*

2 = h(2k

2 - 1) < 2(2

k

2 - 1), where h is a natural number. Hence 1 < h < 2, which is impossible. Hence A is genuine. (ii) Immediate by (i). Note: (FQ) has only one class of integral solutions, which is ambiguous. Theorem 3.5. Let k be a natural number. Then the following are equivalent: (i) 2k

2 — 1 is a square number. (ii) The totality of ambiguous classes of integral solutions of (Fk) consists of a single class. In consequence, if 2k

2 — 1 is not a square number, then every class of integral solutions of (Fk) is genuine. Proof. By using Proposition 3.1 and Theorem 3.4. Theorem 3.6. Consider the Diophantine equation (Fk), where k is a natural number. Let x

n + y

n\/

2, where n = 0,1, 2,..be the sequence of all non-negative integral solutions of x

2 - 2

y

2 = 1. Let X

: + Y*y/2

, (where r — 1,2, . .., m), be the only integral solutions of (Fk) such that: 0 < X; < 2k - 1 and 0 < Y* < k - 1. X

n + y

n\/2 = (x; + Y

r*V2)(x

n + y

nV2) for all n = 0,1,..., < + Y'

nV2 = (X; - Y

r*V2)(x

n + y

nV2) for all n = 1, 2,...,