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 47 (a) Z

n e A and Z'

n £ A for all n > 1 if and only ifY* < k - 1. (

b) Z

n e A for all n > 1 and Z'

n G A for all n > 2 if and only if Y* = k- 1. (ii) Let A be ambiguous, (whence Y* = 0, while 2k

2 -1 is a square number

). Then the only (non-negative) integral solutions X + Yy/2 of () which belong to A and satisfy the inequality Y > k + 1 are all Z

n for every n > 1. Proof, (i) By Theorem 1.2 (iv) we have: Yn+1 > yn > Y'

n >0 for all n > 1, where Y^ = 2X* - 3Y*. (a) Hence, we have Y[ = 2X* - ZY* > k + 1 if and only if (2X*)

2 > (3Y* + k + l)

2, that is if and only if (Y* - (k - l))(Y* + 7k + 5) < 0, and so if and only if Y * < k — 1

. Consequently, by Proposition 4.3, the only (non-negative) integral solution X + Yy/2 of (Fic), which belong to A or A and satisfy the inequality Y > fc + l are all Z

n 6 A and all Z'

n 6 A, n = 1,2,. .., for which Y* < k - 1. (b) Hence, Y

x' = 2X* - 3Y* = k + 1 if and only if Y* = A: - 1. Thus, the only (non-negative) integral solutions X+Y \/2 of (F^

), which belong to A or A and satisfy the inequahty Y > k -f 1 are all Z

n G A for all n > 1 and all Z'

n G 1 for all n > 2 if and only if Y* = jfe - 1. (ii) By Theorem 1.2 (i) the following hold true: Y

n+

l > Y

n > 0 for all n = 0,1,. .while Y* = Y

0 = 0 and Y

x = 2y/2k

2 - 1. Also, (by direct computations) we show that Y\ > A;+ 1. Consequently, the only (non-negative) integral solutions X + Yy/

2 of (F^), which belong to .4 and satisfy the inequahty Y > A: + 1 are all Z

n for every n > 1. Proposition 4.5. Consider the Diophantine equation (Fi). Let X

n+Y

nV

2 EE (1 + 0V2)(3 + 2y/2)

n for all n = 0,1,.... Then the only (non-negative) integral solutions X +Yy/

2 of(F\), such that Y > 2, are all X

n + Y

ny/

2 for every n > 2. Proof. By using Theorem 1.2 (i). Theorem 4.6. Let k be a natural number. Consider the Diophantine equation (F

k). Let Z

; ~ X* + Y*y/2, (where r = 1, 2,. . ., m) be the only integral solutions of (F^) such that: X* > 0 and 0 < Y; < k - 1.