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1 C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 99 FASC. I ON THE DIOPHANTINE EQUATION x y = z BY A. R O T K I E W I C Z WARSZAWA It was shown by Terjanian [] that if is an odd rime and x, y, z are ositive integers such that x y = z then divides x or y. From the theorem of Terjanian the resent author [9] deduced that if x y = z then either 8 3 x or 8 3 y. In [0] the imossibility of the diohantine equation x y = z was established under the conditions x, y =, and either z, z, or z, z rime > 3 [0], Theorem T. In a joint aer with A. Schinzel [] we roved that if x, y, z are ositive integers such that x y = z where is a rime greater than 3 then either x or y, and if x y = z where x, y and z are non-zero integers then < y, x < 8y, which extends Terjanian s result []: if x y = z then either x or y, as well as Chao Ko s result [], [3]: the equation x = z has no solutions in ositive integers if is a rime greater than 3. Here we shall rove the following. Theorem. If x, y =, is an odd rime and x y = z then either x or y, and there exist corime ositive integers α and such that 3 z = α where and α mod, x = α α α..., y = α α α

2 6 A. R OTKIEWICZ P r o o f. Let x y = z. If xy then x y mod, which is imossible. Without loss of generality we can assume that y. We have y = z x and by Theorem T of [0] we have y, hence y. Since x y z = 0, a theorem of Vandiver [6],. 37, Theorem 06 shows that y y mod 3. Since y, 3, we have 3 y, hence y. Now we shall rove that y. We have x = z y, or 5 x = z y z y z y. From y it follows that z y, hence z y z y, z y =. z y z y = e, where e is an odd ositive integer; hence z z y z 3 y mod 8, z y z 3 y mod 8, z y mod 8, y 0 mod 8 and finally y 0 mod. we have y. From x iy x iy = z we obtain x iy = i r a bi, r = 0,,, 3. The factor i r can be absorbed into the th ower, and so we need only consider r = 0. From x, y = it follows that a, b =. 6 x iy = a bi, a, b =, hence 7 x = a 8 iy = a bi a bi a bi... abi, a bi bi. a 3 bi Since x y = a b, y, x, we have ab. From x and 7 it follows that a. b. Since y, 8 gives b. b and since a, b = we have a, =. From 7 we obtain 9 x = a a a 3 b a 5 b... ± From a, b= it follows that a, a a b... ± b =. b.

3 DIOPHANTINE EQUATION 7 0 a = α. From 8 we get y = b a a 3 b... ± 3 a b 3 b and since b, a = we have b, a a 3 b... ± a b 3 b =. 3 From it now follows that there exists a ositive integer such that = b. Since y, and show that b =, hence. 3 b = / where. From 6, 0 and 3 we get x iy = α i, 5 x iy = α i, hence z = α /, and thus 6 z = α. From and 5 we get x = α i α i = α α α i α i y = = i α 3 α 3 3 α..., 5, α and formulas 3 and are roved. By the theorem of Vandiver we have z z mod 3, and since z, = we have z mod 3. Since z = α / and we have z α mod 3, and so 7 α mod.

4 8 A. R OTKIEWICZ This comletes the roof of Theorem. Let z y = z with x, y, z =, 0 < x < y and >. Inkeri in 953 [] showed that if xyz then x > 3 /log3, and if xyz then x > 3 and y > 3. The author in 960 [8] roved that for any natural number n >, x n y n = z n imlies x > 3 n, y > 3 n. Inkeri and van der Poorten in 980 [5] roved that if x y = z with x, y, z =, 0 < x < y and > then z x >. Brindza, Györy and Tijdeman in 985 [] roved that for any natural number n >, if x n y n = z n then x > n n/3. Here we shall rove the following Theorem. If x y = z with x, y, z =, 0 < x < y, > then z >. If x y = z, x, y, z =, 0 < x < y, > then there exist corime ositive integers α and such that z = α /, where 8 3, α mod and z > 3. P r o o f. Let x y = z. By we have z = α > >. Let x y = z, y. By Theorem of [9] we have 8 3 y. From and it follows that = b and from 8 3 y and we get 8 3 b =, hence 8 3 and z = α /, α mod. z > 83 = 8 6 > 6, hence z > 3. This comletes the roof of Theorem. REFERENCES [] B. Brindza, K. Györy and R. Tijdeman, The Fermat equation with olynomial values as base variables, Invent. Math , [] Chao Ko, Acta Sci. Natur. Univ. Szechuanensis 960, [3], On the diohantine equation x = y n, xy 0, Sci. Sinica Ser. A 965, [] K. I n k e r i, Abschätzungen für eventuelle Lösungen der Gleichung im Fermatschen Problem, Ann. Univ. Turku. Ser. A I 6 953, 9. [5] K. Inkeri and A. J. van der Poorten, Some remarks on Fermat s conjecture, Acta Arith , 07. [6] E. Landau, Vorlesungen über Zahlentheorie, Bd. III, Leizig 97; rerint Chelsea, 97. [7] P. Ribenboim, 3 Lectures on Fermat s Last Theorem, Sringer, New York 979. [8] A. Rotkiewicz, Une remarque sur le dernier théorème de Fermat, Mathesis , [9], On Fermat s equation with exonent, Colloq. Math. 5 98, 0 0.

5 DIOPHANTINE EQUATION 9 [0] A. Rotkiewicz, On the equation x y = z, Bull. Acad. Polon. Sci. Sér. Sci. Math ,. [] A. Rotkiewicz and A. Schinzel, On the diohantine equation x y = z, Colloq. Math , [] T. N. Shorey and R. Tijdeman, Exonential diohantine equations, Cambridge University Press, 98. [3] G. Terjanian, Sur l équation x y = z, C. R. Acad. Sci. Paris Sér. A B , [], L équation x y = az et le théorème de Fermat, Séminaire de théorie des nombres de Bordeaux, Année , exosé no. 9. INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES ŚNIADECKICH WARSZAWA, POLAND Reçu ar la Rédaction le