Mathematics of the Jewish Calendar/Gauss' Formula for the Date of Pesach

Gauss

Gauss' Formula for the Date of Pesach

In 1802, the mathematician Carl Friedrich Gauss published a formula to calculate the date of the first day of Pesach in any year. Albert Einstein is said to have remarked of the formula that not only could nobody but Gauss have produced it, but it would never have occurred to anyone but Gauss that such a formula was possible.

Let $\left[{\frac {m}{n}}\right]_{r}$ be the remainder when integer $m$ is divided by integer $n$.

Define:

$A$ $=$ Jewish year

$a=\left[{\frac {12A+17}{19}}\right]_{r}$

$b=\left[{\frac {A}{4}}\right]_{r}$

$M+m=32+{\frac {4343}{98496}}+\left(1+{\frac {272953}{492480}}\right)a+{\frac {b}{4}}-\left({\frac {313}{98496}}\right)A$

where $M$ is the integer part and $m$ the fractional part of the calculated number.

Then 15 Nisan is Mth March in the Julian calendar unless it is postponed for one of the three following reasons:

Define:

$c=\left[{\frac {3A+5b+M+5}{7}}\right]_{r}$

If $c$ = 2 or 4 or 6, then 15 Nisan is ($M$+1)th March.

If $c$ = 1, $a$ > 6 and $m$ ≥ 1367/2160, then 15 Nisan is ($M$+2)th March.

If $c$ = 0, $a$ > 11 and $m$ ≥ 23269/25920, then 15 Nisan is ($M$+1)th March.

It is trivial to adapt the formula to find the dates of the other main festivals; for example, Purim is always 30 days before Pesach and Rosh Hashana is 163 days after Pesach.

Note that the formula gives the date in the Julian calendar, so an adjustment to the Gregorian calendar is necessary. To find the difference, take the year and discard the last two digits; call the result $C$. Calculate

$C-2-\left[{\frac {C}{4}}\right]$

where, in the division, just take the integer part and discard the remainder. Thus the adjustment is to add 11 days for 1700-99, 12 days for 1800-99, 13 days for 1900-2099 and 14 days for 2100-99.

Note that in the very long term, due to calendar drift, it may be hard to know which year of the Julian calendar this Nisan 15th is in. (See Mathematics of the Jewish Calendar/The long-term accuracy of the calendar.)

References

• "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.
• Hamburger, M. (1896) Crelle's Journal, 116, p.90 gave the first rigorous proof.
• Paper by Zvi Har'El, Technion, Haifa, Israel