Happiness Is Integral But Not Rational

Bland, Andre; Cramer, Zoe; Castro, Philip de; Domini, Desiree; Edgar, Tom; Johnson, Devon; Klee, Steven; Koblitz, Joseph; Sundaresan, R.

doi:10.4169/mathhorizons.25.1.8

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Proof Of Lemma 281

Alternative Bases and Domains1

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2021

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“…Unfortunately for the study of fractional base happy numbers, as shown in [6], for any p > q > 1, the only p q -happy number is 1. This is because any other number that would map to 1 by S e, p q is not in Z + .…”

Section: Proof Of Lemma 28mentioning

confidence: 99%

“…In a different direction, happy functions defined using expansions into fractional bases were introduced in [6] and studied further in [56]. Although each positive integer has a unique representation in a given fractional base, it should be noted that, given a fractional base expansion, it is generally not immediately evident whether the number is an integer or some other rational number.…”

Section: Alternative Bases and Domainsmentioning

confidence: 99%

See 1 more Smart Citation

Happy Numbers, Happy Functions, and Their Variations: A Survey

Grundman

Hall-Seelig

2021

La Matematica

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The happy function maps a positive integer (expressed in base 10) to the sum of the squares of its digits. A happy number is a positive integer that, under repeated applications of the happy function, maps to 1. The iterative behaviors of the happy function and its many variations have been the subject of study for over 75 years. In this survey, we discuss the arc of this research, explaining the ideas behind the methods and their general evolution. We explore and organize the wide range of variations, providing full proofs of the key results.

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Section: Proof Of Lemma 28mentioning

confidence: 99%

Section: Alternative Bases and Domainsmentioning

confidence: 99%