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 49 Theorem 4.9. Consider the Diophantine equation (Fk), k = 0,1,.... Let X + YV

2 be a non-negative integral solution of (Fk). Let x = (X -1)/2 and N(x) = x

2 + (x + l)

2. Then N(x) = Y

2 + k

2

. Moreover, the following are equivalent: (i) N(x) is composite, (ii) Y > k + 1. Proof. The equality N(x) — Y

2 -f k

2 follows by direct computations, while the equivalence of (i) and (ii) follows from Theorems 4.2 and 1.1. Theorem 4.10. Let N(x) = x

2 + (x + l)

2. Consider the Diophantine equation (F

k), k = 0,1,.... Let X* + Y*\/2, (where r = 1,2,

..., m) be the only non-negative integral solutions of (F

k) such that: 0 < Y* < k - 1 for k > 1, while, for k = 0 we have: X* = Y* — 1 for all r = 1, 2,..., m. Let X

n + Y

nV2 = (x; + y;%/2)(3 + 2>/2)

n, X^ + y^ - (x; - y

r*v^)(3 + 2>/2)

n for ail n = 0,1,..., (for a typical r). Let x

n = (X

n - l)/2 and x

n = (X

n - l)/2 for every n — 0. 1,. . .. Let R

n, R

n, where n = 0,1,. . be the sequences defined by the recursive formmulae: R

n

+1 = 34

R

n - Ä

n_! - 8(2A;

2 + 1) for all n = 1, 2,..., where Ä

0 = Y/ + A:

2, R

x = (2X

r* + 3Y/)

2 + (for a typical r). R

n+

l = 347^ - R

n_

x - 8(2/c

2 + 1) for all n = 1, 2,. . ., where R'

q = Y*

2 + R[ = (2X* - 3Y

r*)

2 + k

2 (for a typical r). Then the following hold true: (I) Let k = 0. The for every integer n there exists an integer m such that: R

n — R

m __ N(x

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

2,..are ail composite. (ii) Let k — 1, whence X* — 1, Y* = 0 for every r = 1,2,..., m. Then R

n = R

n = N(x

n) for every n > 0.