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Perfect powers in products of terms in an arithmetical progression. (English) Zbl 0708.11021
The authors consider the diophantine equation $(*)\quad m(m+d)... (m+(k-1)d)=by^{\ell}$ in positive integers b, d, k, $$\ell$$, m, y, subject to the conditions: P(b)$$\leq k$$, $$\gcd (m,d)=1$$, $$k>2$$, $$\ell >1$$, $$y>1$$ and $$P(y)>k$$. Further, it is assumed that $$\ell$$ is prime. (N.B. P(x) denotes the greatest prime factor of x.)
A comprehensive overview is given of the present state of knowledge concerning equation (*). Considerable improvements are obtained of recent results of R. Marszalek [Monatsh. Math. 100, 215-222 (1985; Zbl 0582.10011)] and the first author [New advances in transcendence theory, Proc. Symp. Durham 1986, 352-365 (1988; Zbl 0658.10024)]. A sample result, incorporated in the authors’ “main aim of this paper”, is the following:
Let equation (*) be satisfied. If $$\ell \in \{2,3,5\}$$ then k is bounded by an effectively computable number depending only on $$\omega$$ (d). If $$\ell \geq 7$$ then k is bounded by an effectively computable number depending only on $$\ell$$ and $$\omega (d_ 1)$$, where $$d_ 1$$ is the maximal divisor of d such that all prime factors of $$d_ 1$$ are $$\equiv 1(mod \ell).$$
The proofs use a result of J.-H. Evertse [Compos. Math. 47, 289-315 (1982; Zbl 0498.10014)] when $$\ell =3$$ and $$\ell =5$$. Otherwise, the proofs are elementary.
Reviewer: R.J.Stroeker

##### MSC:
 11D61 Exponential Diophantine equations
Full Text:
##### References:
  P. Erdös , Note on the product of consecutiue integers (I) and (II) , J. London Math. Soc. 14 (1939), 194-198 and 245-249. · JFM 65.1145.01|0026.38801  P. Erdös , On a diophantine equation , J. London Math. Soc. 26 (1951), 176-178. · Zbl 0043.04309 · doi:10.1112/jlms/s1-26.3.176  P. Erdös , On the product of consecutive integers III , Indag. Math. 17 (1955), 85-90. · Zbl 0068.03704  P. Erdös and J.L. Selfridge , The product of consecutive integers is never a power , Illinois J. Math. 19 (1975), 292-301. · Zbl 0295.10017  P. Erdös and J. Turk , Products of integers in short intervals , Acta Arith. 44 (1984), 147-174. · Zbl 0497.10033 · eudml:205930  J.-H. Evertse , On the equation axn - byn = c , Compositio Math. 47 (1982), 289-315. · Zbl 0498.10014 · numdam:CM_1982__47_3_289_0 · eudml:89576  D. Hanson , On a theorem of Sylvester and Schur , Canad. Math. Bull. 16 (1973), 195-199. · Zbl 0271.10003 · doi:10.4153/CMB-1973-035-3  H. Iwaniec and J. Pintz , Primes in short intervals , Monatsh. Math. 98 (1984), 115-143. · Zbl 0544.10040 · doi:10.1007/BF01637280 · eudml:178195  M. Langevin , Plus grand facteur premier d’entiers en progression arithmétique , Sém. Delange-Pisot-Poitou, 18e année, 1976-77, no. 3, 7 pp. · Zbl 0373.10006 · numdam:SDPP_1976-1977__18_1_A3_0 · eudml:110958  R. Marszalek , On the product of consecutive elements of an arithmetic progression , Monatsh. Math. 100 (1985), 215-222. · Zbl 0582.10011 · doi:10.1007/BF01299269 · eudml:178250  O. Rigge , Uber ein diophantisches Problem , 9th Congress Math. Scand., Helsingfors, 1938, Mercator, Helsingfors, 1939, 155-160. · Zbl 0021.01003  T.N. Shorey , Perfect powers in values of certain polynomials at integer points , Math. Proc. Camb. Phil. Soc. 99 (1986), 195-207. · Zbl 0598.10029 · doi:10.1017/S0305004100064112  T.N. Shorey , Perfect powers in products of integers from a block of consecutive integers , Acta Arith. 49 (1987), 71-79. · Zbl 0582.10012 · eudml:206071  T.N. Shorey , Some exponential diophantine equations, New Advances in Transcendence Theory , ed. by A. Baker, Cambridge University Press, 1988, pp. 352-365. · Zbl 0658.10024  T.N. Shorey , Some exponential diophantine equations II, Number Theory and Related Topics , Tata Institute of Fundamental Research, Bombay, 1988, pp. 217-229. · Zbl 0748.11022  T.N. Shorey and R. Tijdeman , On the greatest prime factor of an arithmetical progression, A Tribute to Paul Erdös , Cambridge Univ. Press, to appear. · Zbl 0709.11004  T.N. Shorey and R. Tijdeman , Perfect powers in arithmetical progression , J. Madras Univ., Section B, 51 (1988), 173-180. · Zbl 1194.11046  J.J. Sylvester , On arithmetic series , Messenger Math. 21 (1892), 1-19 and 87-120. · JFM 23.0181.02  R. Tijdeman , Applications of the Gel’fond-Baker method to rational number theory , Coll. Math. Jańos Bolyai, 13. Topics in Number Theory (Debrecen, 1974), North-Holland, Amsterdam, 1976, pp. 399-416. · Zbl 0335.10022  R. Tijdeman , Diophantine equation and diophantine approximations, Number Theory and Applications , Kluwer Acad. Publ., Dordrecht, 1989, pp. 215-243. · Zbl 0719.11014  Kunrui Yu , Linear forms in the p-adic logarithms , Acta. Arith. 53 (1989), 107-186. · Zbl 0699.10050 · eudml:206218  Kunrui Yu , Linear forms in p-adic logarithms II , Compositio Math. 74 (1990), 15-113. · Zbl 0723.11034 · numdam:CM_1990__74_1_15_0 · eudml:90010
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