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Journal Article

Pi, the Primes, Periodicities, and Probability

Stephen D. Casey and Brian M. Sadler
The American Mathematical Monthly
Vol. 120, No. 7 (August–September 2013), pp. 594-608
DOI: 10.4169/amer.math.monthly.120.07.594
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Page Count: 15
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Pi, the Primes, Periodicities, and Probability
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Abstract The theory of numbers has repeatedly shown itself to be both practical and beautiful. This paper gives an example of this duality. We present a very efficient (and practical) algorithm for extracting the fundamental period from a set of sparse and noisy observations of a periodic process. The procedure is computationally straightforward, stable with respect to noise, and converges quickly. Its use is justified by a theorem, which shows that for a set of randomly chosen positive integers, the probability that they do not all share a common prime factor approaches one quickly as the cardinality of the set increases. The proof of this theorem rests on a (beautiful) probabilistic interpretation of the Riemann zeta function.

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