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Computers have their limitations, not the least of which is that they are finite machines. Suppose we have a computer that is very finite, able only to use the number symbols 1, 2, 3, 4, and 5, and the operation symbols +, ×, and e, where aeb will mean the number a raised to the bth power. Moreover, our computer cannot form sentences (meaningful strings of numerals and operators) longer than 5 symbols in length. Thus, for instance, our computer can handle “2+4” and “3e5+2,” but not “1+1+1+1.” We'll stipulate the further rules that all operators have equal precedence and are evaluated from left to right, and that any two number symbols must be separated by an operator. Thus, for instance, the number twelve cannot be represented as “12,”, but may be represented as “3×4” or as “2×5+2,” or as “2e2×3.”
Now, we will say that a number is conceivable for our computer if it can be represented using the symbols and rules above, and inconceivable otherwise. For example, it is already shown that 12 is conceivable. Since the number of allowable combinations of the eight symbols into 5-symbol strings is limited, it is clear that most numbers are inconceivable for our computer.
Today's challenge: What are the largest conceivable and the smallest inconceivable numbers?


Solution to yesterday's challenge

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