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Perfect powers in products with terms from arithmetic progression

Vrijeme: 20.1.2010
Predavaonica: 005
Predavač: Kalman Gyory
Naziv: Perfect powers in products with terms from arithmetic progression

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.

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