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##n = p_1^{a_1} \cdot p_2^{a_2} \cdot p_3^{a_3} \cdot \dots##

where ##p_1##, ##p_2##, ##p_3##, etc. are all the primes, and ##a_1##, ##a_2##, ##a_3##, etc. are non-negative integers. (The theorem also essentially states that no two distinct combinations of the latter will produce the same natural number.)

I have been thinking that it's relatively easy to prove that if we lift the restriction of the exponents being non-negative, ie. we allow any integer as exponent, then the statement is true for any positive

*rational*number n. In other words, any positive rational number n can be written as the above type of product (where the exponents are integers).

(The proof is easy to sketch by realizing that in the most-simplified forms of rational numbers the numerator and denominator are relatively prime, ie. they don't share any prime factors, which means the same prime number is never needed for both the numerator and the denominator.)

I am not sure about the uniqueness, however. Could two different combinations of integers produce the same rational number? I think that completeness (ie. any rational number can be written as a finite product like the above) is relatively easy to prove, but I'm not so sure about uniqueness.

Also, I'm not sure how this "extended" version of the theorem could be succinctly expressed. The original theorem can be expressed as "any positive whole number can be written as a unique product of primes", but this extended version is more difficult to express in such a simple manner.