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Assuming that we are considering bases up to 36 (after which we run out of letters), then we can find a unique solution. This Python program runs in 39ms.

Solution:The base is 30.I think the puzzle certainly implies (although it is not explicitly stated), that we are looking at bases up to 36. However, we can calculate the values of the given numbers in bases larger than 36, and we find there are many possible bases in which more than one of the given numbers are prime. If we remove the final line of the program then it will start to find larger bases that give solutions (75, 120, 148, 190, 228, 250, 280, 327, 340, 394, …). And if you replace the call to

is_prime()tois_prime_mr()(a fast Miller-Rabin primality test, the implementation of which was provided by Brian Gladman), then you can find large bases even quicker (e.g. base 250,000 is a solution).By considering the parity of the base we see that ODD and NAB are odd if the base is even (and even if the base is odd), and BIG and BASE are odd if the base is odd (and even if the base is even). PRIME is always even. So we only need to test up to two numbers in each base.