On the Moessner Theorem on Integral Powers

Long, Calvin T.

doi:10.2307/2314178

Cited by 10 publications

(14 citation statements)

References 3 publications

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“…Following Perron's proof of Moessner's theorem, further generalizations were soon made by Ivan Paasche [32] and Hans Salié [40]. Paasche showed how incrementally increasing the gap between the elements dropped in Moessner's sieve, In 1966, Moessner's sieve was picked up by Calvin T. Long [22], who observed that the triangles generated by Moessner's sieve, 1 (2.4)…”

Section: Inductive Contributionsmentioning

confidence: 99%

A Formal Study of Moessner's Sieve

Urbak¹,

Danvy²

2017

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First of all, I would like to express my gratitude towards Olivier Danvy for his invaluable guidance over the last six months along with his ability to see the potential of my abilities, even when I could not. His knowledge and good humor were a vital part in the process. I am also grateful to Ole Østerby for generously sharing his expertise about Horner's method and abstract algebra. Throughout, Markus Wüstenberg has been a wonderful office mate. I also want to thank the designers and developers of the Coq proof assistant, Proof General, Emacs, and L A T E X: these tools have been instrumental here. Coq, in particular, makes it possible for computer scientists to actually do mathematics, as unexpected as that may be. This dissertation is the punch line of six enriching years as student and also as employee in the Department of Computer Science at Aarhus University. It is also the punch line of an upbringing for which I am grateful to my family. Last but not least, I want to thank Sofie for her loving support and our many technical discussions.

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Section: Inductive Contributionsmentioning

confidence: 99%

A Formal Study of Moessner's Sieve

Urbak¹,

Danvy²

2017

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“…As a result, our formalization corresponds well to the paper by Niqui and Rutten, and is very compact. -Lastly, in the process of formalizing Niqui and Rutten's proof, we uncovered a simple proof of Long [10,11] and Salié's [20] generalization. Though (once done) the generalization is not at all complicated, it was surprising to us that the extended version is just a corollary of the original Moessner's Theorem, and that the the bisimulation did not have to be modified.…”

Section: Contributionmentioning

confidence: 99%

“…Long [10,11] and Salié [20] generalized Moessner's result to apply to the situation in which the initial sequence is not the sequence of successive integers (1, 2, 3, . .…”

Section: Long and Salié's Generalizationmentioning

confidence: 99%

Moessner’s Theorem: An Exercise in Coinductive Reasoning in Coq

Krebbers

Parlant

Silva

2016

Theory and Practice of Formal Methods

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Dedicated to Frank de Boer on the occasion of his 60th birthday.Abstract. Moessner's Theorem describes a construction of the sequence of powers (1 n , 2 n , 3 n , . . . ), by repeatedly dropping and summing elements from the sequence of positive natural numbers. The theorem was presented by Moessner in 1951 without a proof and later proved and generalized in several directions. More recently, a coinductive proof of the original theorem was given by Niqui and Rutten. We present a formalization of their proof in the Coq proof assistant. This formalization serves as a non-trivial illustration of the use of coinduction in Coq. During the formalization, we discovered that Long and Salié's generalizations could also be proved using (almost) the same bisimulation.

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“…. Long [5,6] discovered the following alternative procedure and generalization. Consider the figure illustrating the Moessner construction for n = 4 above.…”

Section: Introductionmentioning

confidence: 99%

“…Long [5,6] and Salié [13] also generalized Moessner's result to apply to the situation in which the first sequence is not the sequence of successive integers 1, 2, 3, . .…”

Section: Introductionmentioning

confidence: 99%