By the prime factorization theorem, every integer can be uniquely factored into a product of prime numbers raised to an integer power :
As discussed above, a mixed-radix Cooley Tukey FFT can be used to implement a length DFT using DFTs of length . However, for factors of that are mutually prime (such as and for ), a more efficient prime factor algorithm (PFA), also called the Good-Thomas FFT algorithm, can be used [27,83,36,45,10,86].^{A.4} The Chinese Remainder Theorem is used to re-index either the input or output samples for the PFA.^{A.5}Since the PFA is only applicable to mutually prime factors of , it is ideally combined with a mixed-radix Cooley-Tukey FFT, which works for any integer factors.
It is interesting to note that the PFA actually predates the Cooley-Tukey FFT paper of 1965 [17], with Good's 1958 work on the PFA being cited in that paper [86].
The PFA and Winograd transform [45] are closely related, with the PFA being somewhat faster [9].