Let $n,d,k\geq2,b,y$ and $\ell\geq3$ be positive integers with the greatest prime factor of b not exceeding k. It is proved that the equation $n (n+d) \dotsb (n+(k-1)d)=b y^{\ell}$ has no solution if d exceeds d1, where d1 equals 30 if $\ell =3$; 950 if $\ell =4$; $5\times 10^4$ if $\ell=5$ or 6; 108 if $\ell=7$, 8, 9 or 10; 1015 if $\ell \geq 11$. This confirms a conjecture of Erdős on the above equation for a large number of values of d.

We show that there is no square other than 122 and 7202 such that it can be written as a product of k−1 integers out of k(≥3) consecutive positive integers. We give an extension of a theorem of Sylvester that a product of k consecutive integers each greater than k is divisible by a prime exceeding k.

It is proved that a product of four or more terms of positive integers in arithmetic progression with common difference a prime power is never a square. More general results are given which completely solve (1.1) with gcd(n, d)=1, k≥3 and 1<d≤104.

We study the Diophantine equation

formula here

in integers x, y > 1, n > 2 and q [ges ] 2. Without loss of generality, we suppose that q is prime.

Saradha and Shorey [SS] considered equation (1·1) with x = z2 and proved that it has no solutions in the cases z > 31 or z ∈ {2, 3, 4, 8, 9, 16, 27}.Notice that the proof for z ∈ {2, 4, 8, 16} is elementary, whereas for z ∈ {3, 9, 27} the proof uses a result of Darmon–Merel [DM] which is a generalization of Wiles' result on Fermat's theorem. The purpose of the present paper is to treat completely the remaining cases and to prove the following result.

It has been conjectured that equation

formula here

has only finitely many solutions. We observe that (x, y, n, q)=(3, 11, 5, 2), (7, 20, 4, 2) and (18, 7, 3, 3) are solutions of (1). These are the only known solutions and perhaps (1) has no other solution. Ljunggren [8] proved in 1943 that (1) with q=2 has no solution other than x=3, y=11, n=5 and x=7, y=20, n=4. Shorey and Tijdeman [16] confirmed the conjecture if x is fixed. Let z>1 be an integer. The main purpose of this paper is to show that (1) has no solution if x is restricted to the infinite set of squares z2 with z[ges ]32 and z∈{2, 3, 4, 8, 9, 16, 27}.

It is shown that apart from

there are only finitely many arithmetic progressions with given differences of equal lengths ≥ 2 and with equal products and that they can be effectively determined.