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 \[ x > G\quad \exp ((4-\delta)(\log G)^{1/2}/\log \log G). \]


11N05 Distribution of primes
11J81 Transcendence (general theory)
Full Text: DOI EuDML


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.