A tight lower bound is not known on the number of required additions, although lower bounds have been proved under some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an Ω("N" log "N") lower bound on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for most but not all FFT algorithms). This result, however, applies only to the unnormalized Fourier transform (which is a scaling of a unitary matrix by a factor of formula_19), and does not explain why the Fourier matrix is harder to compute than any other unitary matrix (including the identity matrix) under the same scaling. Pan (1986) proved an Ω("N" log "N") lower bound assuming a bound on a measure of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear. For the case of power-of-two "N", Papadimitriou (1979) argued that the number formula_20 of complex-number additions achieved by Cooley–Tukey algorithms is "optimal" under certain assumptions on the graph of the algorithm (his assumptions imply, among other things, that no additive identities in the roots of unity are exploited). (This argument would imply that at least formula_21 real additions are required, although this is not a tight bound because extra additions are required as part of complex-number multiplications.) Thus far, no published FFT algorithm has achieved fewer than formula_20 complex-number additions (or their equivalent) for power-of-two "N".