Math 55a alum
Scott Kominers
notes that 5777 = Floor(ϕ^{18}), where ϕ is the
“golden ratio”
^{½}) / 2*x*² − *x* − 1

Numerically, ^{18} = 5777.9998269…^{18}^{18}^{18} + ϕ^{−18}
is an integer — namely the 18th
Lucas number *L*_{18} — because
−1/ϕ is the algebraic conjugate 1−ϕ of ϕ,
so ^{18} + ϕ^{−18}^{18}*T*^{18} if *T* is
a linear operator on a two-dimensional space whose eigenvalues are
*T*(*x*,*y*) = (*x*+*y*, *x*)

Now ϕ^{18} + ϕ^{−18} =
(ϕ^{9} − ϕ^{−9})^{2} + 2
= L_{9}^{2} + 2
(in general
_{2m}
= L_{m}^{2} ± 2*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, …*x*^{2} = *x*^{r}^{r}*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* ≡ 0 mod 2^{r},
*x* ≡ 1 mod 5^{r}*x* ≡ 1 mod 2^{r},
*x* ≡ 0 mod 5^{r}^{r−½}*r*-digit*r*=2, this solution is 76,
which is both ^{3/2} = 31.62…*three*-digit

What has all this to do with the Jewish holidays? Well 76,
besides being *L*_{9} and a ^{2}**Z**/19**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 *L*_{18} − 2*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

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

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:

- Scott explains that he found the formula
5777 = Floor(ϕ simply by entering 5777 into the OEIS search window; indeed the second hit is Sequence A014217, whose^{18}) term is*n*-thFloor(ϕ . (The first hit is Sequence A053755, whose^{n}) term is*n*-th4 ; most of the other top hits involve Fibonacci/Lucas/ϕ matters, including*n*^{2}+ 15777 = and the fact that 5777 is the sixth-smallest Lucas pseudoprime.)*F*_{27}/*F*_{9} - 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. - 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. - 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 tolog(1.5) / log(3) ≈ 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). - 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”).