One Is the Loneliest Number But Also a Happy Number?
Consider the number 17...
1^{2}+7^{2}=50 ....
5^{2}+0^{2}=25 ....
2^{2}+5^{2}=29 ....
2^{2}+9^{2}=85 ....
8^{2}+5^{2}=89 ....
8^{2}+9^{2}=145 ....
1^{2}+4^{2}+5^{2}=42 ....
4^{2}+2^{2}=20 ....
2^{2}+0^{2}=4 ....
4^{2}=16 ....
1^{2}+6^{2}=37 ....
3^{2}+7^{2}=58 ....
8^{2}+5^{2}=89 ....
Any cyle under this process (e.g. 89...145...42......37...58) is called a Cheery Sequence. In the case of the number 17, its cycle has 8 terms.
Question 1: Do all positive integers generate Cheery Sequences?
Question 2: Is there a pattern in their periods...that is, do certain types of numbers produce certain cycles and/or periods?
New Information: A Happy Number is an integer whose Cheery Sequence has a period of 1. Why is 10 a Happy Number?
Question 3: Do all integers generate Cheery Sequences with periods of 8 or 1?
Question 4: Do all Happy Numbers have a Cheery Sequence or cycle involving only a repeating 1?
Question 5: Is there an infinite number of Happy Numbers?
Question 6: Are there a pair of consecutive Happy Numbers? Trio of consecutive Happy Numbers in a row? Nconsecutive Happy numbers for any N?
Question 7: Does "Happiness" depend on the base? That is, investigate these same questions in bases other than base 10.
Hint: Just start playing...and answers for the questions will start to develop.
Solution Commentary: Hope your investigations were fun and fruitful, that is, you are Happy and Cheery as well!
If you want to explore the given relationships further, consider the "assumed" original source: Donald Duncan's "Happy Integers," Mathematics Teacher, November 1972, pp. 627629...or even what may be the original source: A. Prges' "A Set With Eight Numbers," American Mathematical Monthly, 1945, pp. 379383.
