let us denote each number as the product of primes in a special way; for example

{2 3 5 7 11 ...}

{2 0 0 1 ...}

represents 28. (line up the primes in the first parentheses with the second one)

notice that in the prime factorization of 28, there a two 2's and one 7.

now, a further observation is that whenever we multiply two numbers, we add the number of each prime in that representation. for example 28 = [2 0 0 1] and 4 = [2 0 0 ...]. when we multiply them, we get 112 which is equal to
[ 4 0 0 1]. similarly, when we raise a number to a power, we multiply each number in that "prime table" by that exponent. for example, 28^2 = [4 0 0 2], which is equal to 2x[2 0 0 1]. to summarize, multiplying numbers implies adding the corresponding prime tables and exponentiation of a number to a power implies multiplying every number in the prime table by the exponent.

now, the question is what addition of numbers translate to in the prime tables. all answers are appreciated.

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