We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of [Formula: see text] and [Formula: see text] are explicitly determined.

Physiological responses of two duckweed species, Lemna gibba and Lemna minor, to hexavalent chromium [Cr (VI)] were studied in axenic cultures using short-term (48 h) treatments by K 2 Cr 2 O 7 (0-200 μM). Chlorophyll (Chl) fluorescence parameters and photosynthetic pigment composition of plants were screened to determine the effects of Cr(VI) exposures. The two duckweed species exhibited different sensitivity in the applied Cr(VI) concentration range. Chl fluorescence parameters of dark-adapted and light-adapted plants and electron transport inducibility were more sensitive to Cr(VI) in L. minor than in L. gibba. We also found fundamental differences in quantum yield of regulated, Y(NPQ), and nonregulated, Y(NO), non-photochemical quenching between the two species. As Cr(VI) concentration increased in the growth medium, L. minor responded with considerable increase of Y(NPQ) with a parallel significant increase of Y(NO). By contrast, in L. gibba only 200 μM Cr(VI) in the growth medium resulted in elevation of Y(NPQ) while Y(NO) remained more or less constant within the regarding Cr(VI) concentration range during 48 h. Photosynthetic pigment content did not change considerably during the short-term Cr(VI) treatment but decrease of Chl a/b and increase of Car/Chl ratios were observed in good accordance with the changes in Chl fluorescence parameters. The data suggest that various duckweed species respond with different sensitivity to the same ambient concentrations of Cr(VI) in the growth medium, and presumably to other environmental stresses too, which may have an influence on their competitive relations when heavy metal pollution occurs in aquatic ecosystem.

Abstract. We propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential diophantine equations. We prove that in a sense the principle is valid for "almost all" equations. Based upon this we propose a general method for the solution of exponential diophantine equations. Using a generalization of a result of Erdős, Pomerance and Schmutz concerning Carmichael's λ function, we can make our search systematic for certain moduli needed in the method.

The complete solution of the Diophantine equation (4m 2 + 1) x + (5m 2 − 1) y = (3m) z Abstract In this short note we give all solutions to the exponential Diophantine equation (4m 2 + 1) x + (5m 2 − 1) y = (3m) z where 20 < m < 90 and m ≡ 3 (mod 6). In view of earlier theorems of Terai, and of Su and Li, this result yields a complete solution set to a problem of Terai.

In this article we develop algorithms for solving the dual problems of approximating linear forms and of simultaneous approximation in number elds F . Using earlier ideas for computing independent units by Buchmann, Peth® and later Pohst we construct sequences of suitable modules in F and special elements β contained in them. The most important ingredient in our methods is the application of the LLL-reduction procedure to the bases of those modules. For LLL-reduced bases we derive improved bounds on the sizes of the basis elements. From those bounds it is quite straight-forward to show that the sequence of coecient vectors (x 1 , ..., x n ) of the presentation of β in the module basis becomes periodic. We can show that the approximations which we obtain are close to being optimal. Thus our algorithm can be considered as such a generalization of the continued fraction algorithm which is periodic on bases of real algebraic number elds.