In any period of HY Hebrew years, what are the minimum and maximum number of months?

In 1802g, the German mathematician Carl Friedrich Gauss published a formula which gave the Julian date of Pesach for any Hebrew year A.

The Gauss Pesach formula appeared without proof of its derivation in “Berechnung des jüdischen Osterfestes”, Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, 5 (1802), 435-437 – reprinted in: Carl Friedrich Gauss Werke (Königlichen Gesellschaft der Wissenschaften, Göttingen, 1874), vol. 6, pp. 80-81.

With a bit of patience, it is actually quite possible to derive the fundamental elements of the Gauss formula.

One of the amazing results is that there are actually 2 formulas that come out of this effort.

The first formula, which was apparently favoured, but not shown, by Gauss, and provides for the maximum number of months in any period of HY years is

where R(12 * Y, 19) is the remainder of 12 * HY divided by 19.

The second formula which can be derived from the method used to derive the Gauss formula giving the minimum number of months is

That second formula can actually be reduced to

When HY MOD 19 = 0 then MINSUM(HY) = MAXSUM(HY)

When HY MOD 19 is not equal to 0 then it can be shown easily that

Correspondent Larry Padwa sent the following correct table driven answer.

Divide the number of years by 19, yielding a quotient Q and remainder R.

Then the minimum and maximum number of years is 235*Q +12*R + X, where X
is obtained from the following table:

R (remainder)    X (minimum)        X (maximum)
-------------    -----------        -----------
      0              0                   0
      1              0                   1
      2              0                   1
      3              1                   2
      4              1                   2
      5              1                   2
      6              2                   3
      7              2                   3
      8              2                   3
      9              3                   4
     10              3                   4
     11              4                   5
     12              4                   5
     13              4                   5
     14              5                   6
     15              5                   6
     16              5                   6
     17              6                   7
     18              6                   7


The first term (235*Q) comes from the fact that every 19 year cycle has
exactly 235 months.

The second term (12*R) comes from the fact that every year has at least
12 months.

The third term (X) reflects the minimum and maximum number of leap years
posssible in the span of years since the last multiple of 19.


Shabbat Shalom,
-Larry

Correspondent Dwight Blevins made a rather intriguing observation leading to the following Weekly Question.

Which are two of the most important arithmetical properties of the 9th year of the mahzor qatan (19 year cycle) known as GUChADZaT?

In 1802g, the German mathematician Carl Friedrich Gauss published a formula which gave the Julian date of Pesach for any Hebrew year A.

The Gauss Pesach formula appeared without proof of its derivation in “Berechnung des jüdischen Osterfestes”, Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, 5 (1802), 435-437 – reprinted in: Carl Friedrich Gauss Werke (Königlichen Gesellschaft der Wissenschaften, Göttingen, 1874), vol. 6, pp. 80-81.

With a bit of patience, it is actually quite possible to derive the fundamental elements of the Gauss formula.

Gauss either established or noticed that when the base formula 12 * A + 5 was divided by 19, the remainder would actually represent the multiple of 19th's of a single lunar month by which the start of the lunar year would exceed the start of the average solar year.

Thus, 12 * A + 5 leaves a remainder of 18 when divided by 19.

Since 18 is the maximum possible remainder after division by 19, that helps to explain exactly why the latest possible Rosh HaShannah's in any given period of time always coincide with the 9th year of GUChADZaT.

For example, Rosh HaShannah 5804H will begin on Mon 5 Oct 2043g. This date, October 5, is in the Gregorian calendar, the latest possible date today for the first day of Tishrei.

Note also that 12 * 5804 + 5 = 69,653 which leaves 18 when divided by 19.

These observations lead to the second major arithmetic property of the 9th year of GUChADZaT. This will be discussed in the next Weekly Question.

What is the second major arithmetic property of the 9th year of the mahzor qatan (19 year cycle) known as GUChADZaT?

The most important part of the Gauss Pesach formula is that it leads to a very simple summation for the number of months that have elapsed up to any year HY in the Hebrew calendar.

Let R(x, 19) = the non-negative remainder after x is divided by 19.

Then,

19 * SUM(HY) = (HY + 2) * 235 + R(12 * HY + 5, 19)

Let XY = the number of years elapsed from some HY

Then,

19 * SUM(HY + XY) = (HY + 2 + XY) * 235 + R(12 * HY + 5 + 12 * XY, 19)

The number of months between HY and HY + XY

= 19 * SUM(HY + XY) - 19 * SUM(XY)
= (HY + 2 + XY) * 235 + R(12 * HY + 5 + XY, 19) - (HY + 2) * 235 - R(12 * HY + 5, 19)
= XY * 235 + R(12 * HY + 5 + 12 * XY) - R(12 * HY + 5, 19)

When HY = 9, then R(12 * HY + 5, 19) = 18

Hence, the number of months in XY years beginning at the 9th year of GUChADZaT

= (XY * 235 + R(12 * XY + 18, 19)- 18) / 19

This single valued expression also represents the minimum number of months that any period of XY years may have.

Therefore, a second property of the 9th year of GUChADZaT is that any period of years beginning in the 9th year will always have the minimum number of months possible for that period of years.

Rabbi Erwin Schild, Rabbi Emeritus of Congregation Adath Israel in Downsview, Ontario, asked the following question.

The following terms need explaining before asking the question.

A parshah or sedrah is the portion of the Mosaic scriptures (Torah) selected for reading during a particular week.

Shabbat Shuvah is the Shabbat preceding the day of Yom Kippur.

Parshah or Sedrah Ha'azinu is found in D'vorim (Deuteronomy) 32:1-52.

Rabbi Erwin Schild, Rabbi Emeritus of Congregation Adath Israel in Downsview, Ontario, asked this week's question.

The following terms need explaining.

A parshah or sedrah is the portion of the Mosaic scriptures (Torah) selected for reading during a particular week.

Shabbat Shuvah is the Shabbat preceding the day of Yom Kippur.

Parshah or Sedrah Ha'azinu is found in D'vorim (Deuteronomy) 32:1-52.

One of the more prevalent practices, among the Jewish people, is that of reading the entire Mosaic text over the course of one Hebrew year. At Simchat Torah, the last few verses are read, and then the entire cycle is repeated once again from Bereshit (Genesis).

The scriptural readings are divided into contiguous weekly portions, which are read in their entirety each Shabbat morning. Each division is known as a Parshah or Sedrah. These portions are arranged so as to be completely read over the course of one Hebrew year.

Each portion is given a special name. The two portions that can each be read on Shabbat Shuvah are Vayelech and Ha'azinu.

Since there are 14 ways of laying out the Hebrew years (14 keviyyot), there exist only 14 ways of dividing the annual Torah reading cycle. As a result, the 14 different divisions can be easily tabulated in very compact form. One such tabulation may be found at the back of certain editions of the Humash (Pentateuch) as translated by Alexander Harkavy, and published by the Hebrew Publishing Co. in New York (1928).

That source shows that the portion Ha'azinu is read whenever Rosh Hashannah begins on either Thursday or Shabbat.