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 ....

42 Fanayiotis G. Tsangaris to that of solving each one of the Diophantine equations (F^). Theorem 1.3. Consider the Diophantine equations (E

) and (FThen the following hold true: (z)i Let (x,y,z) be an integral solution of (E

) with y > z. Let X = 2x -f 1 and Y = 2y — (k — 1),. where k = y — z. Then X + Yy/2 is an integral solution of (F^

). (02 If y í 0, -1 and z ^ 0, -1 then \Y\ ^ k ± 1. (u)x Let X + Y yj

2 be an integral solution of (F

tt). Let (2.1) x = (X - l)/2, y = (Y + k - l)/2 and z = (Y - k - l)/2. Then (x,y,z

) is an integral solution of(E). (n)

2 If\Y\ f k ± 1, then y ± 0,-1 and 2 / 0,-1. Proof. (i)i By direct computation. (i)2 Clear because |Y| = k ± 1 implies (y — 0, —1) or (z = 0, —1). (ii)i Let X -f Yyj

2 be an integral solution of (Fk). Then it is easily proved by parity considerations that the numbers (2.1) are integers. Also X = 2x + 1, y = 2y - (k - 1) and k = y - z, whence (F^) implies (2x + l)

2 - 2(2y - (y-z- l))

2 = 2

(y - zf - 1, that is x(x + 1) = y(y + 1) + z(z + 1). (ii)2 Is proved in a way similar to the proof of (i)

2, namely (y = 0, —

1) or (z = 0. -1) imply |F| = k± 1. Note. The transformation leading from (E

) to (F

*.) emanate from GAUSS (Art. 216 in [1]) 3. Determination of all integral solutions of the equation X

2 - 2Y

2 = 2k

2 - 1, where k = 0,1,... Proposition 3.1 is crucial for the location of the fundamental solutions of (FFC). Further, Theorem 3.4 characterizes the classes of solutions of (Fk), (as regards genuiness or ambiguity) in terms of their representing fundamental solutions. Special attention is given to the case of 2k

2 — 1 being a square