Powersums representing residues mod p^k, from Fermat to Waring

The ring Z_k(+,.) mod p^k with prime power modulus (prime p>2) is analysed. Its cyclic group G_k of units has order (p-1)p^{k-1}, and all p-th power n^p residues form a subgroup F_k with |F_k|=|G_k|/p. The subgroup of order p-1, the core A_k of G_k, extends Fermat's Small Theorem (FST) to mod p^{k>1}, consisting of p-1 residues with n^p = n mod p^k. The concept of "carry", e.g. n' in FST extension n^{p-1} = n'p+1 mod p^2, is crucial in expanding residue arithmetic to integers, and to allow analysis of divisors of 0 mod p^k. . . . . For large enough k \geq K_p (critical precison K_p < p depends on p), all nonzero pairsums of core residues are shown to be distinct, upto commutation. The known FLT case_1 is related to this, and the set F_k + F_k mod p^k of p-th power pairsums is shown to cover half of units group G_k. -- Yielding main result: each residue mod p^k is the sum of at most four p-th power residues. Moreover, some results on the generative power (mod p^{k>2}) of divisors of p^2-1 are derived. -- [Publ.: "Computers and Mathematics with Applications", V39 N7-8 (Apr.2000) p253-261]Comment: (9 pgs) Publ.: "Computers and Mathematics with Applications", V39 N7-8 (Apr.2000) p253-261. See http://www.iae.nl/users/benschop/nw-abstr.htm - Intro at http://www.iae.nl/users/benschop/fewago.htm -- See also http://arXiv.org/abs/math.GM/0103067 (on primitive roots

Propagation velocity and rate of attenuation of surface waves on a homogeneously fluidized bed

Surface waves with a frequency of 0.5¿2.5 Hz were generated on a homogeneously fluidized bed. The propagation velocity and rate of attenuation of the induced pressure fluctuations were measured using signal averaging techniques. The measured wave velocity and attenuation rate correlated well with predictions based on a theory which considers the bed as an incompressible liquid with low viscosity. From the rate of attenuation an effective bed viscosity was calculated between 1.2 and 6.0 Pa · s. At high frequencies the wave generator produced high-amplitude density waves