The Personalities of Numbers [part 2 of 3]

Some numbers might be Evil or Odious but others are Happy. A Happy number can be described like this:

happy-numberIf you iterate the process of summing the squares of the decimal digits of a number, then it is easy to see that you either reach the cycle 4→16→37→58→89→145→42→20→4 or arrive at 1. In the latter case you started from a happy number.”

The description comes from a paper in the Rocky Mountain Journal of Mathematics Volume 30, Number 2 (2000), pp. 565-570, entitled: ‘On Happy Numbers’ by professor Samir Siksek of the Mathematics Institute, University of Warwick, UK, and Esam El-Sedy [possibly a.k.a. Essam S. El-Sedy of the Ain Shams University  Department of Mathematics]

“Several questions” say the team “are asked about happy numbers, including: ‘How many consecutive happy numbers can you have? Can there be arbitrarily many?’ It is the purpose of this paper to show that there are sequences of consecutive happy numbers of arbitrary length.” The paper makes progress towards the answers, and can be read in full here.

Note that if a number doesn’t satisfy the conditions described above, it must, lamentably, be an Unhappy Number, (a.k.a. a Sad Number) like 33.

Coming soon : Weird numbers