Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0708.11021
Shorey, T.N.; Tijdeman, R.
Perfect powers in products of terms in an arithmetical progression. (English)
[J] Compos. Math. 75, No.3, 307-344 (1990). ISSN 0010-437X; ISSN 1570-5846/e

The authors consider the diophantine equation $$ (*)\quad m(m+d)... (m+(k-1)d)=by\sp{\ell} $$ in positive integers b, d, k, $\ell$, m, y, subject to the conditions: P(b)$\le k$, $\gcd (m,d)=1$, $k>2$, $\ell >1$, $y>1$ and $P(y)>k$. Further, it is assumed that $\ell$ is prime. (N.B. P(x) denotes the greatest prime factor of x.) \par A comprehensive overview is given of the present state of knowledge concerning equation (*). Considerable improvements are obtained of recent results of {\it 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: \par 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 $\ell \ge 7$ then k is bounded by an effectively computable number depending only on $\ell$ and $\omega (d\sb 1)$, where $d\sb 1$ is the maximal divisor of d such that all prime factors of $d\sb 1$ are $\equiv 1(mod \ell).$ \par The proofs use a result of {\it J.-H. Evertse} [Compos. Math. 47, 289-315 (1982; Zbl 0498.10014)] when $\ell =3$ and $\ell =5$. Otherwise, the proofs are elementary.
[R.J.Stroeker]

MSC 2000:: *11D61 Exponential diophantine equations

Keywords: exponential diophantine equation; greatest prime factor

Citations: Zbl 0582.10011; Zbl 0658.10024; Zbl 0498.10014

Cited in: Zbl 0777.11007 Zbl 0763.11014 Zbl 0763.11015

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]