## On the Oesterlé-Masser conjecture.(English)Zbl 0597.10042

For any positive integers x, y and z define $$G=G(x,y,z)$$ to be the product of the primes which divide xyz. Oesterlé asked if there exists a constant $$c_ 1$$ such that for all positive integers x, y and z with $$(x,y,z)=1$$ and $$x\geq y\geq z$$ we have $$x<G^{c_ 1}$$. Masser later conjectured that for any positive real number $$\epsilon$$, $$x<c_ 2(\epsilon)G^{1+\epsilon}$$ where $$c_ 2(\epsilon)$$ is a positive number which depends on $$\epsilon$$ only.
In this article we prove, with the above hypotheses, that log x$$<c_ 3 G^{15}$$ where $$c_ 3$$ is an effectively computable constant. Further, let $$\delta$$ be a positive real number. We show that there exist infinitely many positive integers x, y and z such that $$x=y+z$$, $$(x,y,z)=1$$ and $x > G\quad \exp ((4-\delta)(\log G)^{1/2}/\log \log G).$

### MSC:

 11N05 Distribution of primes 11J81 Transcendence (general theory)

### Keywords:

prime numbers; linear forms in logarithms
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### References:

 [1] Ennola, V.: On numbers with small prime divisors. Ann. Acad. Sci. Fenn. Series AI440, 1-16 (1969). · Zbl 0174.33903 [2] Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math.73, 349-366 (1983). · Zbl 0588.14026 · doi:10.1007/BF01388432 [3] Hall, Jr., M.: The diophantine equationx 3?y 2=k. In: Computers in Number Theory. A.O.L. Atkin and B.J. Birch (eds.). Proc. Sci. Res. Council Atlas Symp. No. 2, Oxford 1969, pp. 173-198. London: Academic Press. 1971. [4] Masser, D. W.: Open problems. Proc. Symp. Analytic Number Th. W. W. L. Chen (ed.). London: Imperial College 1985. [5] Pillai, S. S.: On the equation 2 x ?3 y =2 X +3 Y . Bull. Calcutta Math. Soc.37, 15-20 (1945). · Zbl 0063.06245 [6] Van der Poorten, A. J.: Linear forms in logarithms in thep-adic case. In: Transcendence Theory: Advances and Applications. A. Baker (ed.), pp. 29-57. London: Academic Press. 1977. [7] Barkley Rosser, J.: Then-th prime is greater thann logn. Proc. London Math. Soc. (2)45, 21-44 (1939). · Zbl 0019.39401 · doi:10.1112/plms/s2-45.1.21 [8] Tijdeman, R.: On the equation of Catalan. Acta Arith29, 197-209 (1976). [9] De Weger, B. M. M.: Solving exponential diophantine equations using lattice basis reduction algorithms. Report Math Inst. University of Leiden. 1986, No. 13. · Zbl 0625.10013
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