Authors’ introduction: A long-standing conjecture claims that the Diophantine equation \[ \frac{x^n-1} {x-1}= y^q, \quad\text{in integers }x>1,\;y>1,\;n>2,\;q\geq 2, \tag{1} \] has finitely many solutions, and, maybe, only those given by \[ \frac {3^5-1} {3-1}= 11^2, \qquad \frac {7^4-1} {7-1}= 20^2 \quad\text{and}\quad \frac {18^3-1} {18-1}= 7^3. \] Among the known results, let us mention that Ljunggren solved (1) completely when \(q=2\), and that Nagell treated the cases \(3| n\) and \(4| n\). For more information and, in particular, for finiteness type results under some extra hypotheses, we refer the reader to T. N. Shorey and R. Tijdeman [Math. Scand. 39, 5–18 (1976; Zbl 0341.10017); Exponential Diophantine equations, Cambridge University Press, Cambridge (1986; Zbl 0606.10011)] and to the recent survey of T. N. Shorey [Exponential Diophantine equations involving products of consecutive integers and related equations. Number Theory, Trends Math., Birkhäuser, Basel, 463–495 (2000; Zbl 0958.11026)].
In the present work, we solve (1) completely when \(x\) is any integer in the interval \([2,10]\) and, under some restriction, when \(x\) is a power of a prime number. In these cases, it is already known that the number of solutions is finite, and that they are bounded by an effectively computable constant. As in the paper [Y. Bugeaud, M. Mignotte, Y. Roy and T. N. Shorey, Math. Proc. Camb. Philos. Soc. 127, 353–372 (1999; Zbl 0940.11020)], we combine several methods in Diophantine approximation together with some computer calculations. Our main tool is a new lower bound for linear forms in two \(p\)-adic logarithms [see Y. Bugeaud, Math. Proc. Camb. Philos. Soc. 127, 373–381 (1999; Zbl 0940.11019)], which applies very well to (1) and allows us to reduce the time of computation considerably.
The main result is the following. Theorem . Let \(t\geq 1\) be a rational integer. Let \(z\) be an integer in the interval \([2,10]\) and let \(p\geq 11\) be a prime number. Then equation (1) has only two solutions \((x,y,n,q)\) with \(x=z^t\), namely \((x,y,n,q)= (3,11,5,2)\) and \((7,20,4,2)\). Further, equation (1) has no solution \((x,y,n,q)\) with \(x=p^t\) and \(\gcd (q,p-1)=1\).
We also solve an old open problem in proving that any number \(x= 11 \dots 11\), with all digits equal to 1 in base 10, is not a pure power, except for \(x=1\).
Note that the proofs of the results obtained in [M. Le, Acta Arith. 69, 91–98 (1995; Zbl 0819.11012)] and in [L. Yu and M. Le, Acta Arith. 73, 363–366 (1995; Zbl 0834.11015)] are incorrect. Indeed, they all depend on Lemma 3 of M. Le (loc. cit.), which is false; see the comment in [P.-Z. Yuan, Acta Arith. 83, 199 (1998)]. In the present work, we supply a correct proof of some of their claims. However, we remark that our method is essentially different from theirs.