Strings of 14 and 15 Consecutive Happy Numbers

In Spring 2012, one of my senior seminar students, Daniel Lyons, tackled the problem of the least strings of 14 and 15 consecutive happy numbers.  His basic idea is to exploit the fact that one can order the digits, which allows the calculations to be 7 million times more efficient than the methods I had used.  

14 in a row :   7888.(1604938271577 nines).1.(345696 nines).3

15 in a row:   N=77.(2222222222222220 nines).3.(97388 nines).3       

Here is Dan Lyon's paper (docx, pdf).  This paper was published in Involve: A Journal of Mathematics, Vol. 6 (2013), no. 4, pp 461-466

Here are worksheets Dan used for the calculations. 

        14 in a row (mw)
        14 in a row after the carry (mw)
        14 in a row before the carry (mw)
        15 in a row after carry 16 digits (mw)
        15 in a row after carry 17 digits  A (mw)
        15 in a row after carry 17 digits PC (mw)
        15 in a row after carry 18 digits  (mw)
        15 in a row after carry 18 digits part 2 (mw)
        15 in a row after carry 18 digits part 3 (mw)
        15 in a row after carry 18 digits part 4 (mw)
        15 in a row before carry 16 digits (mw)
        15 in a row matching 7 digits before, 8 digits after (mw)

 

Styer Home Page           04/17/2015