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matt grime
Jan30-04, 08:04 AM
Sci Advisor
HW Helper
P: 9,396
Of course it can.

Define the 'arithmetic' in anyway you choose that happens to agree with that specific example on those particular numbers. Eg, the Collatz rules except when a_n=1,000,000,000, and anything but division by two there.

If you mean is there another way of defining a Collatz type relation which agrees on ALL values of a_n, then the answer is yes and no. There are equivalent constructions, but functionally they are the same. By that I mean

sin(x) and cos(x-pi/2) are functionally the same, but have different 'labels'