## Christmas, Hanukkah, and Fibonacci

January 4, 2012

Happy New Year everyone!  As usual I’m way overdue adding to this blog, and intend to write a proper post shortly on one of the subjects that have recently piqued my interest.  However, in the meantime, I can’t resist quickly pointing out something which came to my notice today, courtesy of my daily email from arXiv; a quirky little article that neatly ties together 3 of the things I have written about recently: calendars, religion, and number sequences (in particular, Fibonacci numbers).

The paper in question is a very short one, attributed to one Shalosh B. Ekhad.  A quick search reveals that this “person” is in fact a computer belonging to a mathematician called Doron Zeilberger, who is quite well known in mathematical circles for his love of computers, and for not being entirely serious all of the time.  However, I will humour him by writing as if it was indeed the entity named Shalosh B. Ekhad who wrote this article.

So, Ekhad became interested in how often the Jewish holiday of Hanukkah coincides with Christmas, and began to run computer searches (which I suppose equates to just thinking about it, if you are actually a computer).    For the non-Jews reading this (I had to look it up myself) Hanukkah is an 8-day holiday which begins on the 25th day of the month of Kislev in the Hebrew Calendar.   The Hebrew calendar is an example of a lunisolar system, in that it takes into account the relative motions of both the sun and the moon. Each year consists of 12 lunar months of either 29 or 30 days, apart from leap years which have an extra month.   Leap years occur 7 times every 19 years; if we think of the first year of a 19-year cycle as being year 1, then the leap years are years 3, 6, 8, 11, 14, 17, and 19.

As you might expect, this means that dates of the Hebrew calendar vary quite wildly in relation to those of our strictly solar Gregorian calendar.  In particular it means that the pattern of years in which Christmas falls within the Hanukkah period is highly unpredictable.  What Ekhad found was that it will happen in 27% of the years of this (3rd) millenium, with this figure falling for subsequent milleniums, until the 9th millenium, when it will stop happening altogether until at least the year 20000 AD.  So far, so mildly diverting.  Much more interesting is the following observation: during the period when the gaps between  between the years 1801 and 7390, that is, in the time-span in which the gaps between years in which Christmas falls within Hanukkah are relatively small, the number of years making up these gaps are always Fibonacci numbers!  (I talked about Fibonacci numbers in this post).  In particular they are always either 2, 3, 5 or 8.  Ekhad then goes on to point out that exactly the same phenomenon occurs for years in which Christmas falls within Sukkot, another Jewish holiday lasting 7 days.

This seems quite incredible…perhaps slightly less so for those who are used to Fibonacci numbers cropping up in the most unexpected places, but it cries out for explanation all the same.  So what is going on here?  Well, the regular occurrence of gaps of 2 and 3 years between these special Christmas-in-Hanukkah years surely has something to do with the fact that the number of non-leap years between leap years in the Hebrew calendar is always either 2 or 3.  The sequence of gaps between leap years (3,2,3,3,3,2,3) would also go some way to explaining the occurrence of 5 and 8 as well, as both of these numbers can be made from sums of consecutive numbers from this sequence.  But then so can 6,9,10,13….

So there’s more to it than that.  Any insights?

One problem mathematicians constantly struggle with is that there are just not enough letters in the world: we long ago exhausted the entire Roman and Greek alphabets (and even some of the Hebrew one), and as a result many letters are used in a bewildering number of different contexts. Well, I will straight away put your troubled minds to rest by stating that, continuing my occasional series on important mathematical constants (am I allowed to call two posts a series?), when I say “phi”($\phi$) I mean the number$\phi ,$otherwise known as the Golden Ratio, or the Divine Proportion. But first, here are some numbers:
$1,1,2,3,5,8,13,21,34,55,89,144,...$