We give all normal integral bases for the simplest cubic field $$L_n$$ generated by roots of Shanks’ polynomial when these exist, that is, $$L_n/{\mathbb {Q}}$$ is tamely ramified. Furthermore, as an application result, we explicit relation between and Gaussian periods in case ramified, which a generalization work Lehmer, Châtelet Lazarus conductor equal to $$n^2+3n+9$$ .

We disprove various claims made by Ramanujan in his, until very recently, unpublished manuscript [1] on the partition and tau functions. Furthermore, the second part of a related paper by G.K. Stanley [17] is corrected (the first part of which was earlier corrected by D. Shanks [16]).

In 1989, Shanks introduced the NUCOMP algorithm [10] for computing the reduced composite of two positive definite binary quadratic forms of discriminant ∆. Essentially by applying reduction before composing the two forms, the intermediate operands are reduced from size O(∆) to O(∆) in most cases and at worst to O(∆). Shanks made use of this to extend the capabilities of his hand-held calculator...

Abstract This paper examines a number of extrapolation and acceleration methods introduces few modifications the standard Shanks transformation that deal with general sequences. One goals is to lay out framework encompasses most known strategies. The also considers Anderson Acceleration (AA) method under new light exploits connection quasi-Newton in order establish local linear convergence resu...

Newell & Shanks (N&S) argue that when awareness measures are more reliable and valid, greater evidence of awareness of supposedly unconscious influences is revealed. A related issue is that unconsciousness is typically the null hypothesis that evidence of awareness will not emerge. As it is difficult to conclude the null, it is also difficult to conclude a lack of conscious awareness.

A new tendency is emerging in cognitive science that pays attention to the possible commonalities between elementary processes of learning (such as classical or instrumental conditioning) and more complex inductive processes (like categorization or causality judgements) (see, e.g., Estes, 1985; Gluck & Bower, 1988; Holland et al., 1986; Pearce, 1988; Shanks & Dickinson, 1987; Schlimmer & Grange...

This paper continues the work begun by D. Shanks and myself in [1] where certain cubic recurrences were used to give a very strong primality test. A complete characterization of the pseudoprimes for this test is given in terms of the periods of the corresponding sequences. Then these results are used to produce various types of pseudoprimes. A discussion of open problems is included.