### 5777 and all that

October 3, 2016 was the Jewish New Year, first of Tishrei 5777. It is unusually late (as will be the remaining Jewish holidays this year), for reasons that relate to some neat arithmetical facts about 5777 and its neighbors 5778 and 5776.

Math 55a alum Scott Kominers notes that $5777 = \lfloor\varphi^{18}\rfloor$, where $\varphi$ is the “golden ratio” $\frac12(1 + \sqrt5) = 1.61803\ldots$, which is the positive root of $x^2-x-1$. (The number $18$ is regarded as auspicious in Judaism, being the gematria value of Hebrew CHAI חי “life, living”.)

Numerically, $\varphi^{18} = 5777.9998269\ldots$, so the floor function is $5777$ only barely; the difference $5778 - \varphi^{18}$ is exactly $1 / \varphi^{18}$. The sum $\varphi^{18} + \varphi^{-18}$ is an integer — namely the 18th Lucas number $L_{18}$ — because $-1/\varphi$ is the algebraic conjugate $1-\varphi$ of $\varphi$, so $\varphi^{18} + \varphi^{-18}$ is the sum of $\varphi^{18}$ and its algebraic conjugate (and thus the trace of $T^{18}$ if $T$ is a linear operator on a two-dimensional space whose eigenvalues are $\varphi$ and $-1/\varphi$, such as $T(x,y) = (x+y,x)$ ).

Now $\varphi^{18} + \varphi^{-18} = (\varphi^9 - \varphi^{-9})^2 + 2 = L_9^2 + 2$ (in general $L_{2m} = L_m^2 \pm 2$, the sign depending on the parity of $m$); and indeed the recently departed year was numbered $5776$ which is a perfect square, indeed the only four-digit square that ends with its own root. Some context for this: For each $r = 1, 2, 3, \ldots$ there are four “idempotents” (solutions of $x^2 \equiv x \bmod 10^r$): the trivial 0 and 1, and two nontrivial solutions that sum to $1 \bmod 10^r$ (in general $x$ is an idempotent in some ring if and only if $1-x$ is an idempotent). The two nontrivial idempotents can be recovered from the simultaneous (“Chinese remainder”) congruences $x \equiv 0 \bmod 2^r, \ x \equiv 1 \bmod 5^r$ or $x \equiv 1 \bmod 2^r, \ x \equiv 0 \bmod 5^r$. At least one of them exceeds $10^{r-\frac12}$ and thus has a $2r$-digit square. In our case of $r=2$, this solution is 76, which is both $0 \bmod 4$ and $1 \bmod 25$. The complementary idempotent $25$ is not another example becuase it is a bit under $10^{3/2} = 31.62 \ldots$, so instead is the unique root of a three-digit square (namely 625) that ends with its own root.

What has all this to do with the Jewish holidays? Well 76, besides being $L_9$ and a mod-100 idempotent, is also $4 \cdot 19$, which makes $5776 = 76^2$ a multiple of 19. (It had to be, because 19 is prime and the Legendre symbol $(5/19)$ is $+1$, so $\varphi$ and $-1/\varphi$ have representatives mod 19 in the “prime field” ${\bf Z} / 19{\bf Z}$ — namely $-4$ and $5$, as it happens — so their 18th powers are 1 by Fermat’s Little Theorem, whence their sum $L_{18}$ is $2 \bmod 19$, and $L_{18} - 2$ is the square of $L_9$.)

Now 18 is auspicious in Judaism, but 19 is a key number in the Jewish calendar, which has lunar months (the Hebrew ירח can be read as either YERACH [month] or YAREACH [moon(*)]), but must not stray too far from the solar year because some of the Jewish festivals are tied to the seasons (e.g. Shavuot is a harvest festival, and during the week of Sukkot one is to eat outside in a makeshift booth, which is tolerable this time of the year but would be problematic in mid-winter). But a lunar month is about 29½ days long, and a solar year about 365¼, longer than 12 lunar months but shorter than 13. So a “lunisolar calendar” must combine years of 12 and 13 lunar months to keep synchronized on average with the Earth’s orbit around the Sun.

(*) Indeed the English word “month” is likewise related with “moon”, and for that matter “moon” itself is sometimes used for “month” as in “honeymoon” and “many moons ago”.
What proportion of years must have a 13th month to maintain this synchronization? About 0.3683, using the lengths of the month and year derived from the values in this Wikipedia page. That is more accuracy than was available thousands of years ago, but the ancients observed the Sun and Moon long and accurately enough to find the excellent approximation 7/19 = 0.3684… to this ratio, meaning that 7 of every 19 years need a thirteenth month. (Such an extra month is nowadays called a “leap month”, making the year a “leap year”; in Hebrew a 13-month year is known somewhat quaintly as a “pregnant year” [שנה מעוברת = SHANAH MəUBERET].) So, to know whether Year $N$ was a leap year you need to know $N \bmod 19$. For those $N$, The Jewish New Year and other Tishrei holidays (Yom Kippur, Sukkot, Shmini Atzeret) come late in the solar calendar on the following year, $N+1$.

Of course we need to also know which 7 years in each cycle of 19 get the extra month. They are spread as evenly as possible over the cycle, which specifies the pattern uniquely, but only up to translation mod 19. The Babylonians chose years 3, 6, 8, 11, 14, 17, and 19 of each 19-year cycle, and the Jewish calendar retains this choice (known as the Metonic cycle, though it was already known in Babylonia and China when Meton of Athens found the 7/19 approximation). So usually leap years are spaced three apart, but twice in each cycle they come closer, and then the next year starts even later than usual. Since 5776 is a multiple of 19, we’ve just had one of these close pairs (17/19), so Rosh Ha-Shanah and the rest are pushed even further back.

The “Metonic cycle” affects not just the solar date but also the start and end time of the Jewish holidays, because they are celebrated from sundown to sundown, so for example Rosh Ha-Shanah 5777 actually started at sundown the evening of the last day of 5776. (This already appears in the first chapter of Exodus, where each day of Creation concludes with the formula VAYHI-EREV VAYHI-VOKER “and there was evening and there was morning”, in that order; EREV is thus used for the “Eve” of a holiday, as in EREV SHABBAT = Friday evening. Note that “New Year’s Eve” = December 31 works the same way, and likewise for other “Eves”.) In this part of the year, and in our Northern Hemisphere, later in the calendar means earlier sundowns. We are near the equinox (Sep.22), when the days are getting shorter most quickly. This also means that each sundown-to-sundown interval is a bit less than 24 hours. The shortest possible Yom Kippur fast would be when the day falls exactly on the equinox; afterwards the fasts get longer as the days keep getting shorter. But this effect is negligible: even at the equinox the interval between sundowns is not as much as 2 minutes less than 24 hours, and the difference between that minimal interval and what we have this year is measured in seconds.

Further notes:
1. Scott explains that he found the formula $5777 = \lfloor\varphi^{18}\rfloor$ simply by entering 5777 into the OEIS search window; indeed the second hit is Sequence A014217, whose n-th term is $\lfloor\varphi^n\rfloor$. (The first hit is Sequence A053755, whose n-th term is $4n^2+1$; most of the other top hits involve Fibonacci/Lucas/$\varphi$ matters, including $5777 = F_{27} / F_9$ and the fact that $5777$ is the sixth-smallest Lucas pseudoprime.) [Added in 2017: also related is $5779 = L_{27}/L_9$.]
2. The Hebrew calendar actually inserts the 13th month in the middle of the year: instead of the sixth month Adar it’s Adar Aleph and Adar Beth. [Adar used to be the last month; the Biblical name for the 1 Tishrei holiday is YOM TRUAH (“day of the [shofar] blast”), not Rosh Ha-Shanah (New Year, literally “head of the year”).] That means that Passover (7th month) and all later holidays are already delayed in the leap year itself. So is Purim, which falls in Adar, and is celebrated on Adar Beth on leap years, though the corresponding day in Adar Aleph is still recognized as “Purim Katan” (“Little Purim”) and marked by minor observances.
3. Since a lunar month is about 29½ days, the months of the Hebrew calendar usually alternate between 30 and 29 days. But the second and third months, normally 29+30, sometimes go 29+29 or 30+30, either to compensate for the discrepancy between 29½ and the actual astronomical period of about 29.53 days, or to prevent ritually awkward coincidences (such as Yom Kippur falling the day before or after Shabbat). So there are not two but six possible lengths of a Jewish year: 353/354/355 for a common year, 383/384/385 for a leap year. See for instance Wikipedia’s extensive page on the Hebrew calendar. It is sometimes claimed that the alignment of the Jewish and the common (Gregorian) calendar, and/or the Jewish calendar and the week, repeats exactly every 19 years, so that your 19th, 38th, 57th, … Jewish birthday always coincides with the common birthday, and/or falls on the same weekday; but this is a misconception: there is neither a condition nor a consequence of the (complicated) rules of the Jewish calendar that forces such a coincidence. For example, 19 years ago Rosh Ha-Shanah fell on Thursday, October 2, while this year’s was Monday the 3rd.
4. The pattern of 3- and 2-year intervals between leap years is the same as the pattern of whole and half notes in a major scale: starting from C/0, two long gaps (C-D-E/0-3-6) and a short one (E-F/6-8), three long gaps (F-G-A-B/8-11-14-17) and a short one (B-C/17-19). That makes sense because 19/7 is 1 more than the ratio 12/7 of the lengths of the chromatic and diatonic scales, which in turn arises as an approximation to $\log(2) \, / \log(1.5)$. So our solar-lunar discrepancy of about 0.3684 is close to $\log(1.5) \, / \log(3) \approx 0.3691$. That numerical approximation is surely pure coincidence, though it does afford a nice mnemonic for the leap-year pattern (which I noticed independently, but see now is also mentioned on that Wikipedia page).
5. The title “5777 and all that” of this page is a parody of the title of Joe Harris and David Eisenbud’s “3264 & All That: Intersection Theory in Algebraic Geometry”, which itself parodies the title “1066 and All That: A Memorable History of England, comprising all the parts you can remember, including 103 Good Things, 5 Bad Kings and 2 Genuine Dates” of the Sellar-Yeatman book described by its Wikipedia entry as “a tongue-in-cheek reworking of the history of England.” As the same Wikipedia entry explains, “3264 refers to the number of smooth conic plane curves tangent to 5 given general conics, an answer to a problem in enumerative geometry.” Note that in the present page 5777 refers to both a mathematical number (as in “3264”) and a year number (as in “1066”).