A celebrated theorem of Erdos and Selfridge states

that the product of consecutive positive integers

is never a power. It is an old conjecture that even the equation

(*) m(m+d)...(m+(k-1)d)=y^n

has no solution in positive integers m, d, k, y, n, with

gcd(m,d)=1, k>=3, n>=2, and (k,n) not equal to (3,2).

This equation has been investigated by many people.

In last ten years the conjecture was confirmed for k<=11

(Gy, Hajdu, Saradha, k=4,5; Bennett, Bruin, Gy, Hajdu, 6<=k<=11).

Recently these results have been extended to the case k<35

by Hajdu, Pinter and the speaker. As in the earlier proofs,

we reduced (*) to systems of ternary equations. However,

our proof required fundamentally new ideas. For k>11, a great

number of new ternary rquations arose that we solved by

combining the modular method with local and cyclotomic considerations.

Furthermore, the number of systems of equations grows so rapidly

with k that, in contrast with the previous proofs, it was practically

impossible to handle the different cases in the earlier way.

We algorithmized our proof which enabled us to emply a computer.

We appplied an efficient, iterated combination of our procedure for

solving the arising new ternary equations with several sieves based on

the ternary equations already solved. Our algorithm seems to work

for larger k as well, but there is of course a computational time limit.