Problem 60783. Convert integers from base 10 to proper primary notation

> However, proper primary notation has the smallest number of digits. So, while 5 can be expressed as pp, pxx, and pxxm, the proper primary notation is pp.

I think this isn't enough to ensure uniqueness: 12 could be pxpp (7 + 3 + 2), or ppxx (7 + 5), for instance.

And this also affects the test cases: -28734 and 153488 in particular, for which my WIP solution finds different proper primary notations, namely 'mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmpmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm' and 'ppppppppppppppppppppppppppppppppppppppppppppxpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppmppppp' respectively.

Thanks Christian. You're right that the current description doesn't ensure uniqueness. How about, "If m, x, and p, are switched to 0, 1, and 2, respectively, for positive n and 2, 1, and 0 for negative n, then the resulting base-10 number would be as small as possible."? Is there a less clumsy way to say that?

@Chris Yes. If you add this requirement, then you can in fact drop the first one, and just write (say) this:

"Because integers can be expressed as sums (and differences) of primes in several ways, several primary notations for a number can exist. To ensure uniqueness, we require that when 1 is added to the factors of the primes, (so that 'p' is replaced with 2, 'x' with 1 and 'm' with 0 in the primary notation), the resulting decimal number is minimal.

For instance, 5 can be represented as pp, pxx and pxmm; since 22 is smaller than 211 or 2100, pp is the proper primary representation of 5. The number 12 can be represented as pxpp and ppxx; since 2122 is less than 2211, pxpp is the proper primary representation of 12."

OK, thanks. I changed the description.

If I understand correctly, for n=-1800 the sorting rule m<x<p should give 'mmmmmmmmmmmmmmmmmmmmmmmmpmxmmmmm' rather than 'mmmmmmmmmmmmmmmmmxmmmmmmmmmmmmmp'; the test suite seems to use m<x<p for positive n, but p<x<m for negative n, so that the results for +n and -n always look the same but with m and p swapped.

I am passing all tests except test 11 (-28734). I am getting a solution of:
'mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmpmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm' which is a smaller integer than your solution:
'mmmmmmmxmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmxmm'.

I changed the problem to deal with non-negative numbers.

Representing negative numbers by swapping all the m's and p's would make for a rather elegant feature of this system. Even so, I don't think the system as a whole will catch on sufficiently to supplant the decimal system, as old and tired as that may be.