We give a more practical variant of Shanks’ 1954 algorithm for computing the continued fraction of logb a, for integers a > b > 1, using the floor and ceiling functions and an integer parameter c > 1. The variant, when repeated for a few values of c = 10, enables one to guess if logb a is rational and to find approximately r partial quotients. 1. Shanks’ algorithm In his article [1], Shanks gav...

1. The authors use a polyexponential equation (equation 1) to describe the time course of the plasma concentrations of a drug in lieu of a compartmental interpretation suggested by Sheiner et al. As a consequence, the effect compartment can now contain any amount of a drug and is not limited to being “negligibly small” as postulated by Sheiner et al. 2. Sheiner et al. introduced keo as the firs...

An identity for the finite sum ∑N 1 z qn−r is given. Related sums (or series) were studied by Scherk, Clausen, Ramanujan, Shanks, Andrews, and others. We use such identity to give new formulas for ∑∞ 1 z qn−r , the Riemann zeta function and the Euler–Mascheroni constant. An irrationality result is also proved.

1. The authors use a polyexponential equation (equation 1) to describe the time course of the plasma concentrations of a drug in lieu of a compartmental interpretation suggested by Sheiner et al. As a consequence, the effect compartment can now contain any amount of a drug and is not limited to being “negligibly small” as postulated by Sheiner et al. 2. Sheiner et al. introduced keo as the firs...

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Let u(n) = f(gn), where g > 1 is integer and f(X) ∈ Z[X] is non-constant and has no multiple roots. We use the theory of S-unit equations as well as bounds for character sums to obtain a lower bound on the number of distinct fields among Q( √ u(n)) for n ∈ {M + 1, . . . ,M + N}. Fields of this type include the Shanks fields and their generalizations.

Metabolism of Anticancer Drugs: a Rapid and Specific Enzyme Immunoassay for Methotrexate NIGEL G. L. HARDING and A N N E P. SIMMONDS Department of Pathological Biochemistry, Royal Infirmary, Glasgow G4 OSF, Scotland, U. K. and GEORGE F. SHANKS Central Institute of Technology, Wellington, New Zealand and JOHN M. WHITELEY Department of Biochemistry, Scripps Clinic and Research Foundation, La Joll...

We describe an algorithm due to Gauss, Shanks and Lagarias that, given a non-square integer D 0; 1 mod 4 and the factorization of D, computes the structure of the 2-Sylow subgroup of the class group of the quadratic order of discriminant D in random polynomial time in log jDj.

This paper proves the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. The result uses ingredients of the proof of the infinitude of Carmichael numbers, along with zero-density estimates for Hecke L-functions.

IN the past seven years four cases of cough fracture with eight fractures were found at Killadeas Hospital and the Chest Clinic, Erne Hospital, Enniskillen. No previous record of cough fracture of a rib in Northern Ireland is known to the writer. Yet any patient with a history of cough followed by pleuritic pain may be suffering from one. Other conditions predisposing to fracture may also be pr...