On the Divisors of the Discriminant of the Period Equation

Lehmer, Emma

doi:10.2307/2373534

Cited by 11 publications

(10 citation statements)

References 5 publications

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“…The author showed in [3] that these primes have the Fibonacci roots 0 = (1 ± v / 5)/2 as quintic residues. In our case, by (3) are quintic residues of p. The discriminant of the period equation [4] becomes in this case…”

Section: Absolute Artiadsmentioning

confidence: 85%

An indeterminate in number theory

Lehmer

1989

J Aust Math Soc A

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Section: Absolute Artiadsmentioning

confidence: 85%

An indeterminate in number theory

Lehmer

1989

J Aust Math Soc A

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“…We can now write the polynomial F*,(t) given in [6] in terms of n as follows: The discriminant of F5(t), given in [7], reduces in this case to…”

Section: Notationmentioning

confidence: 99%

“…Since there were no known quintic units, while the cyclotomic quintic polynomial [6] and its discriminant [7] were given by the author many years ago, it seemed worthwhile to try to discover some quintic units as linear transforms of the periods for some sequence of primes. This was accomplished for primes of the form p = n4 + 5n3 + 15n2 + 25n + 25.…”

Section: Introductionmentioning

confidence: 99%

Connection Between Gaussian Periods and Cyclic Units

Lehmer¹

1988

Mathematics of Computation

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Abstract.This paper finds that all known parametric families of units in real quadratic, cubic, quartic and sextic fields with prime conductor are linear combinations of Gaussian periods and exhibits these combinations. This approach is used to find new units in the real quintic field for prime conductors p = n* + 5n3 + 15n2 + 25n + 25.1. Introduction.The idea that it might be of interest to explore the connection between Gaussian periods and cyclic units arose from the obvious fact that for 4p = L2 + 27 Shanks's "simplest cubic" [10] and the Gaussian cubic are related by a translation. Moreover, this is also the case for p = a2 + 16 for Marie Gras's "simplest quartic" [2] and the cyclotomic quartic, as well as the "simplest quadratic"

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“…For example, they can be used for deriving the coefficients of the period equation (Lehmer [13]) and its discriminant (Lehmer [14]), and for determining certain quintic residuacity conditions (Lehmer [12], Williams [26], [27], [28]). We now know that they can be efficiently computed.…”

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On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants

Buchmann¹,

Williams²

1987

Math. Comp.

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Abstract. Let if be any totally complex quartic field. Two algorithms are described for determining whether or not any given ideal in if is principal. One of these algorithms is very efficient in practice, but its complexity is difficult to analyze; the other algorithm is computationally more elaborate but, in this case, a complexity analysis can be provided.These ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime p = 1 (mod 5). Given any quadratic (or quintic) nonresidue of p, it is shown that these cyclotomic numbers can be efficiently computed in 0((\ogp)3) binary operations.

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On the Divisors of the Discriminant of the Period Equation

Cited by 11 publications

References 5 publications

An indeterminate in number theory

An indeterminate in number theory

Connection Between Gaussian Periods and Cyclic Units

On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants

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