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 ....

50 Fanayiotis G. Tsangaris Moreover, the numbers R

2, R3, • • are all composite. (iii) Let k > 2 and Y

r* = 0 Then R

n = R

n = N(x

n

) for every n > 0. Moreover, the numbers R\, R

2,

..are all composite. (iv) Let k >2 and Y

r* = k - 1. Then = iV(í

n) and R

n — N(x

n) for every n > 0. Moreover, the numbers Ri, R

2,

. . and also the numbers R

2, R

3,

..., are all composite. (v) Let k>2 and 0 < V

r* < k - 1. Then I I R

n — N(x

n

) and R

n = N(x

n) for every n > 0. Moreover, the numbers Ri, i?2, • • and aiso the numbers R

1, R

2,

. . are all composite. Note: For the cases (iv) and (v) we have: I R

m ^ R

n for anj m, n. Proof, (i) The unique class of integral solutions of (FQ) is ambiguous. By Theorem 2.4 in [5] and Corollary 4.8 we have: + YNV^ EE 6N+L + V2n+

= (1 + + y„v/

2) = (1 + >/2)

2"

+

1 for all n = 0,1,.... Hence, by the definition of ambiguous class and Theorem 1.3, for every integer n there exists an integer rn such that: R

n = R

m = N(x

n), where x

n = (bn+i ~ l)/2. According to Corollary 4.8, the only (non-negative) integral solutions X -f YV

2 of (i'o) such that Y > 1 are all Y

n

+1 = T]

2

n+

3 for every n > 0. Hence by Theorem 4.9, the numbers Ri,R

2, . .. are all composite. (ii) Obviously X* = 1, Y* = 0 for every r = 1, 2,..., m because k = 1. Hence, R

n = R'

n for all n = 0,1, — Now, Theorem 1.3 implies Rn = N{x

n) = Y

n

2 4- k

2 = Y

n

2 + 1 for all n > 0.