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 41 B

r. Moreover, X

n, Y

n,X

n and Y

n are determined by the following recursive formulae: 2x\X

n - X

n_i for n = 1,2,... X*, X\ = X\X* + dyiY* and r = 1,2,.. ., m. 2x

1Y

n - y

n_ 1 for n — 1,2,... Y*, Y\ = y\X* + XiY* and r - 1, 2,..., m. 2x

lX

n - X-i for n = 1,2,. . . I X*

, X

x — x\X* — dy{Y* and r - 1,2,..., m. for n = 1, 2,... ^ = yi^; - and r = l,2,...,m. Theorem 1.3. Consider the Diophantine equation (F), C ^ 0. Let X* + Y*y/d be the fundamental solution of a class A

r of integral solutions of (F). Let x\ + y\Vd be the fundamental solutions of (P) and X

n + Y

nVd = (x; + Y

r*Vd)(x

, + V

l Vd)

n = (X; + Y

r*Vd)(x

n + y

nVd), x'

n + Y^Vd = (x; - Y;y/d)(

X

l + yiVd)

n for all n = 0,1,.... Let R

n = Y

2 + k

2 and R

n = Y-f k

2

, where k is a fixed integer. Then the numbers R

n and R

n are determined by the following recursive formulae

: R

n+i = 2x

2R

n - R

n_! - 2

k

2(x

2 - 1) + 2

y\C, where R

0 = Y*

2 + k

2 and R

x = (y

xX; + x

xY

r')

2 + k

2

. R'

n+

1 = 2x

2R

n - R

n_! - 2

k

2(x

2 - 1) + 2

y\C, where R

0 = Y

r*

2 + k

2 and R[ = (yi X* - XiY*)

2 + k

2

. 2. Reduction of the Diophantine equation x(x -f 1) = y(y -f 1) + z(z + 1) to a family of Fermat equations Theorem 2.1 below aims at reducing the problem of solving the Diophantine equation (E) x(x + 1) = y(y + 1) + z(z + 1) (1.3) ^n+1 with A^o ^n+l with Y

0 (1.4) 1 I with Xn Y 71+1 with Yn =