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 ....

48 Fanayiotis G. Tsangaris Let A

r be the corresponding classes of integral solutions of (Fk) with fundamental solutions Z*. Let Z

n = X

n + Y

nV2 EE (X; + Y;V2){3 + 2\/2)

N for all n = 0,1,..., Z'

n = X'

n + Yy

2 EE (X; - Y

r*V2)(Z + 2V2)

n for all n = 1,2,.... for an (arbitrary

) typical r. Then the only (non-negative

) integral solutions X + Y yj

2 of (Fk), which satisfy the inequality Y > k + 1, are the following: (i) All Z

n £ A

r and all Z'

n £ A

r for every n > 1 if and only if 0 < Y* < k- 1. (ii) All Z

n £ A

t for every n > 1 and all Z

n £ A

R for every n > 2 if and only if 0 < Y* = k - 1. (Hi) All Z

n £ A

r for every n > 1 if and only if Y* = 0 for k > 2. (iv) All Z

n £ A

r for every n > 2 if and only if Y; = 0 for k = 1. Proof. By using Propositions 4.4, 4.5 and Theorem 3.6. Theorem 4.7. Let k be a natural number. Consider the Diophantine equation (Fk). Let X* + Y* y/2, (where r — 1,2,... ,m) be the only integral solutions of (Fk) such that: X; > 0 and 0 < Y

r* < k - 1. Let X

n + Y

nV2 EE (x; + Y;%/2)(3 + 2^2)

n for all n = 0,1,... and r = 1, 2,. .., m, x'

n + Y'

ny/2 = (x; - Y;V2)( 3 + 2v/2)

n for all n = 1,2,... and r - 1, 2,..., m. Then the only (non-negative) integral solutions X + YV

2 of (Fk) such that Y > k + l are the following: (i) All X

n + y

n

V/2 and all X'

n + (with n > 1) for every Y

r* with 0 < Y; < k - 1, when k > 2. (ii) All X

n + Y

nV

2 (with n > 1) and all X'

n + Y^V

2 (with w > 2j for 0 < y

r* = k - 1, when Jfe > 2. (iii) All X

n + y

nV2 (with n> 1) for Y

r* = 0, when /c > 2. fivj AÜ X

n + Fnv^ (with n > 2) for Y* = 0, when k = 1. Proof. By using Theorems 3.6 and 4.6. By Corollary 3.7 it follows that Corollary 4.8. The only non-negative integral solutions X -f YsJ

2 of (FQ) such that Y > 1 are: + Y

n\f~2 for every n = 1,2,