[gizmo686]

> I conjecture that for every j > 0 in R, a number n exists so that any two n x n matrices can be multiplied together in O(n^(2+j)) steps.

Is this stated correctly? Because it seems almost meaningless as stated. You start with "for every j, there exists an n such that...". That would mean that for the rest of the statement, n and j are constant. So you are just saying that you can multiply constant sized matrices in constant time. Technically true, but I feel like you are trying to claim something stronger.

[JohnKemeny]

It should simply say:

for any j>0 there exists an algorithm multiplying nxn matrices in time O(n^{2+j}).

[Lucasoato]

You are correct, I apologize for the confusion! :)

[hnfong]

Assuming your claim is not equivalent to "matrix multiplication is in O(n^2)", then it is false assuming conventional mathematics (i.e. uncountable sets exist), because j in R is uncountable, and algorithms are countable...

[ernesth]

You do not need a different algorithm for each real. Just take the algorithm that proves the statement for rational p. It proves the statement for all reals bigger than p. (hence having as many algorithms as there are rational (countably many) is enough)

[hnfong]

Then that’s basically saying there is a O(n^2) algorithm which I dealt with in my first sentence.

[akoboldfrying]

Are algorithms on a Real RAM [1] countable? That would mean that there are constants you can't express.

[1] https://en.wikipedia.org/wiki/Real_RAM