Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

FREJMAN, D. and GRYTCZUK, A., On a problem of W. Sierpinski

On a problem of W. Sierpinski 99 If ß is an even number then we have 2ß

2 +3 = 3 (mod 4) and a

2 = 1 (mod 4) and similarly we obtain a contradiction mod 4. Thus we obtain that the équation (9) has no solution in natural numbers Q, ß. Now we can consider the case d — 3. By (5) and (4) it follows that (10) i^i

1) =4'

5' (^•

B) = 1From (10) we obtain (11) A = a

2, B = 3

2, x

2-l = 3aß

, (a,ß) = l. By (11) and (6) it follows that (12) 3a

4 - 2ß

2 = 1. Applying Bumby's result (see Lemma) to (12) we have (13) < a,ß > = < 1,1 >,< 3,11 > . Hí a = ß = 1 then A = B = 1 and x

2 - 1 = 3 thus x = 2. From (5) we have 2y + 1 = 3 • 1 thus y — 1, which contrardicts to our assumpiton y > 1. Hence a = 3, ß = 11 and we obtain A = a

2 = 3

2 = 9, B = ß

2 = 11

2 = 121, x

2 - 1 = daß = 3

2 • 11 = 99 thus x = 10. From (5) 2y + 1 = dA = 3 • a

2 = 3 • 3

2 = 27, thus y = 13. Because = y -f 1, thus 2: = 14. Hence < x

:y,z > = < 10,13,14 > and the proof of the Theorem is complété. References [1] R. T. BUMBY, The Diophantine équation 3x

4 -2y

2 = 1, Math. Scand., 21 (1967), 144-148. [2] W. SIERPINSKI, Elementary Theory of Numbers, PWN Warszawa, (1987).