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 ....

52 Fanayiotis G. Tsangaris Let y > z. Let also k = y - z, X = 2x + 1 and Y = 2y - (k - 1). Then, according to Theorem 4.2 (ii), X-\-Y \/2 is a (non-negative) integral solution of (F

k), with Y > k + 1. Hence, X + Y y/\

2 is a solution of type (i) or (ii) or (iii) or (iv) of Theorem 4.7 or it is a solution X + Y^2 of (F

0) with y > 1 (see Corollary 4.8). Also, N(x) = Y

2 + k

2

. Hence, by Theorem 1.3 N(x) is equal to some R

n or some R

n. Finally, the appropriate index n for which N(x) = R

n or N{x) = R

n is obtained by applying Theorem 4.6 to the respective case as in (i)-(v) of Theorem 4.10. This ends the proof of the Theorem. Theorem 4.12. (Determination of all composites of the form N(x) = x

2 + (x -f l)

2) Consider the Diophantine equations (F

k

) X

2 - 2y

2 = 2k

2 - 1, where k = 0,1,.... Let X: + Y;V2, (where r = 1, 2,. .., m), be the only non-negative integral solutions of (Fk) such that: 0 < Y* < k - 1 for k > 1, While, for k = 0 we have: X* = Y

r* = 1 for all r = 1, 2,..., m. Let R

n, R

n be the sequences defined by the recursive formulae: R

n

+1 = 34

R

n - R

n_ 1 - 8(2

k

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

0 = Y/ + k

2

, R

x = (2X* + 3y/)

2 + k

2 (for a typical r). R

n+

l = 34

R

n - R

n_i - 8(2k

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

q = Y

r** + k

2

, R[ = (2X*

r - 3Y

r*)

2 + k

2 (for a typical r). Then, the only composite numbers of the form N(x) = x

2 -f (x + l)

2 are the following: (i) R

UR

2

i... (for k = 0). (ii) Ä

2, Ä3, • • • (for k = 1 and Y

r* = 0j. (iii) R

u R

2,

... (for k >2 and Y* = o). (iv) Äi

, Ä

2,

. .. together with R

? (for k > 2 and Y

r* = k - 1). (v) Ri, • • • together with R

l

: R

2, • • • (for k > 2 and for all Y* such that 0 < y; < k - 1). Proof. By using Theorems 4.10 and 4.11. Theorem 4.13. (Sieve-algon thm for the determination of all primes of the form N(x) = x

2 -f (a: -f l)

2 in an Interval [.

5,M

], where M is a (positive) integer)