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Number Theory Generalized Erdos-Moser conjecture

mathbalarka

Well-known member
MHB Math Helper
Mar 22, 2013
573
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)
 
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