[FOM] arithmetic without product

[Cédric Doucet]

Could you provide where Sylvester's argument was published please?

2018-07-21 10:20 GMT+02:00 José Manuel Rodriguez Caballero <
josephcmac at gmail.com>:

> Consider the following theorem:
>
> For any positive integer k, there is a positive integer n having at least
>> k representations as sum of consecutive positive integers.
>
>
> For example, for k = 3 we have n = 9 = 4+5 = 2+3+4.
>
> For the statement of this theorem we need the addition. If we also have
> the multiplication, then our theorem can be proved using a well-known
> argument due to J. J. Sylvester. On the other hand, if we have not
> multiplication and we restrict ourselves to first order logic, the
> existence of a proof of this result is not so clear. Is there an
> "arithmetic with just addition" where this theorem can be stated but it
> cannot be proved?
>
> Kind Regards,
> Jose M.
>
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