Generalized Erdos-Moser conjecture

[mathbalarka]

I have been researching on a generalization of Erdos-Moser, which asks for ordered tuple of consecutive integers with first $n-1$ integers, summed and exponentiation by $n$, equals the $n$-th power of the last and the greatest. The generalization can be observed as

$$3^2 + 4^2 = 5^2$$
$$3^3 + 4^3 + 5^3 = 6^3$$

Note that similar patterns doesn't work for higher powers, neither does any other examples. I have found a bit of reference on Dickson, which says Escott proved the impossibility for 4 and 5. I haven't tried out anything for now, but just looking for resources.

Some seems to refer this as Cyprian's theorem or Cyprian's last thereon, erroneously. The works referring that as a name is mostly done by Kochanski and doesn't seem to come up with anything worthwhile.

I have found also a few OEIS entries, but nothing is said there except the usual sequence. the best I can find is the Escott reference, which I have to look for whether it is available online or not.

Please post away here if anyone finds anything else.

(Editorial Note : I have written this in a hurry so apologies if it seems a bit scrambled, fortunately I will have some time later and edit it then. I will also try to find some other resource and edit those in too)

 

[Aaron Bex]

You may be interested in looking into the work of Andrzej Schinzel and his "generalized cyclotomic polynomials". These polynomials are related to the problem you are studying. In particular, Schinzel's Generalized Riemann Hypothesis states that all zeros of these polynomials are real. This would imply that there are no such ordered tuples of consecutive integers as you describe.

You may also want to look into the works of Michael Filaseta, who has done some research on this problem. In particular, he has shown that if there exists a tuple $(a_1, \dots, a_n)$ satisfying the condition
$$\sum_{i=1}^{n-1} a_i^n = a_n^n,$$
then $a_n < 2^n$.