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 45 (for a typical r

). Then the following hold true: (i) Let Y* > 0 and k > 2. (Case of genuine classes of integral solutions of (F

k)). Then the pairs {X

n,Y

n) and {X'

n,Y^) are determined by (1.3) and (1.4) (for x

x = 3, y

l = 2 and d = 2). (ii) Let Y; = 0. (Case of ambiguous classes). Then the pairs (X

n

, Y

n

) are determined by (1.3). Moreover, in case (i) all pairs (X

n

, Y

n) together with all pairs (X'

n, Y'

n) constitute the set of all non-negative integral solutions of (F

k) which belong to the class with typical fundamental solution X* + Y/V^. Also

, in case (ii) all pairs (X

n, Y

n) constitute the set of a

11 non-negative integral solutions of (JFk) which belong to the class with typical fundamental solution X*-\-Q\/2. Proof. By using Theorems 3.4, 3.5, 1.2(vi) and Proposition 3.1. Corollary 3.7. The sequence of a

11 positive integral solutions (X

n

, Y

n) of (F

0) is determined by (1.2) (for X

n = f

2

n+i,Y

n = V

2

n+i,fi = 1,6 = 7, tji = 1 and r/

3 = 5). 4. Determination of all prime and composite numbers of the form x

2 + (x + l)

2. In Theorem 4.2 it is shown that every positive (integral) solution of (T) leads to a non-negative solution of a certain (F

k) and vice-versa. Theorems 4.6, 4.7 together with Corollary 4.8 determine all (F

k) whose non-negative solutions (taken together) lead to all positive solutions of (T). In Theorem 1.1a primality criterion is given for numbers of the form N(x) = x

2 +(x-\-i)

2

. Composite numbers of the form N(x) are characterized (in terms of a suitable solution of (F

k)) in Theorem 4.9. The recursive determination of all composite numbers of the form N(x) is given by Theorems 4.10, 4.11 and 4.12. This leads to our final Theorem 4.13, which constitutes an algorithm (sieve) for the determination of all primes of the form N(x). Lemma 4.1. Let X + Yy/2 be a non-negative integral solution of (Fk). Let x = (X - l)/2, y = (Y + k - l)/2 and ZE (Y - k - l)/2. Then x, y, z are natural numbers if and only if Y > k + 1. Proof. Easy and so omitted. Theorem 4.2. Consider the Diophantine equations (F

k) and (T). Then the following hold true: