Oral Presentations

Title

Ecstatic Happy Numbers

Document Type

Oral Presentation

SURC 202

16-5-2013

16-5-2013

Abstract

The focus of the project is to determine the density of Happy Numbers in bases 2 ≤ b ≤ 10 based on what we know about happy numbers in general. A natural number n is called happy if the sequence n, ƒ(n), ƒ(ƒ(n)), ƒ(ƒ(ƒ(n))),…, eventually reaches 1, where ƒ(n) denotes the sum of the squared digits of n. A natural number in base 2 ≤ b ≤ 9 is happy if we sum the squares of the digits and convert the number back to base b prior to the next iteration. Repeated iterations were performed using PARI-GP developed source code. For integer bases 2 ≤ b ≤ 10, there exist infinite sequences of numbers that are happy in all subsequent bases from 2 to b. Generalizations of happy numbers in consecutive ordinal bases are formalized. The set of ecstatic numbers in base b ≥ 2 is a subset of the set of happy numbers. More than half of all natural numbers are ecstatic base 2. This is evident from the observation that all even natural numbers are ecstatic base 2 as well as some odd numbers. All numbers that are ecstatic in bases b ≥ 4 are base 10 odd. Since all numbers are base 4 happy, there are no base 3 ecstatic numbers. This result follows from the definition of an ecstatic number as being the largest base for which a number is happy in all preceding bases.

Faculty Mentor(s)

Jane Whitmire

Mathematics

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May 16th, 1:10 PM May 16th, 1:30 PM

Ecstatic Happy Numbers

SURC 202

The focus of the project is to determine the density of Happy Numbers in bases 2 ≤ b ≤ 10 based on what we know about happy numbers in general. A natural number n is called happy if the sequence n, ƒ(n), ƒ(ƒ(n)), ƒ(ƒ(ƒ(n))),…, eventually reaches 1, where ƒ(n) denotes the sum of the squared digits of n. A natural number in base 2 ≤ b ≤ 9 is happy if we sum the squares of the digits and convert the number back to base b prior to the next iteration. Repeated iterations were performed using PARI-GP developed source code. For integer bases 2 ≤ b ≤ 10, there exist infinite sequences of numbers that are happy in all subsequent bases from 2 to b. Generalizations of happy numbers in consecutive ordinal bases are formalized. The set of ecstatic numbers in base b ≥ 2 is a subset of the set of happy numbers. More than half of all natural numbers are ecstatic base 2. This is evident from the observation that all even natural numbers are ecstatic base 2 as well as some odd numbers. All numbers that are ecstatic in bases b ≥ 4 are base 10 odd. Since all numbers are base 4 happy, there are no base 3 ecstatic numbers. This result follows from the definition of an ecstatic number as being the largest base for which a number is happy in all preceding bases.