In exact fraction of year, how much longer is the Hebrew year over the Gregorian year?

In days, the difference of the average Hebrew year over the average Gregorian year is easily found to be the exact fraction
**(6939d + 16h + 595p) / 19 - 146097d / 400 = 10,643d / 2,462,400**.

In **years**, the exact fraction is a bit more complicated, since it can only be given either as fraction of **Hebrew year** or as fraction of **Gregorian year**.

Due to the fact that both the **Hebrew years** and the **Gregorian years** are of different lengths, the correspondence between the Hebrew and Gregorian calendars repeats itself in a cycle of

**14,389,970,112 Hebrew years**
which is also

Let **H** the number of Hebrew years **(hy)** which exactly equal **G** gregorian years **(gy)**.

It is interesting to note that when reduced, all of the above fractions have the numeratorThen H * hy = G * gy Hence, hy - gy = hy - H * hy / G = hy * (G - H) / G = hy * 170,288 / 14,390,140,400 = 10,643 / 899,383,775 Hebrew years = gy * G / H - gy = gy * (G - H) / H = gy * 170,288 / 14,389,970,112 = 10,643 / 899,373,132 Gregorian years

Exactly how many years are required for a one year difference between the number of Hebrew and Julian years?

In exact fraction of year, how much longer is the Hebrew year over the Gregorian year?

In days, the difference of the average Hebrew year over the average Julian year is easily found to be the exact fraction
**(6939d + 16h + 595p) / 19 - 1461d / 4 = -313d / 98,496**.

In **years**, the exact fraction is a bit more complicated, since it can only be given either as fraction of **Hebrew year** or as fraction of **Gregorian year**.

Due to the fact that both the **Hebrew years** and the **Julian years** are of different lengths, the correspondence between the Hebrew and Julian calendars repeats itself in a cycle of

**1,007,318,592 Hebrew years**
which is also

Let **H** the number of Hebrew years **(hy)** which exactly equal **J** Julian years **(jy)**.

It is interesting to note that when reduced, all of the above fractions have the numeratorThen H * hy = J * jy Hence, hy - jy = hy - H * hy / J = hy * (J - H) / J = hy * -8,764 / 1,007,309,828 = -313 / 35,975,351 Hebrew years = jy * J / H - jy = jy * (J - H) / H = jy * -8,764 / 1,007,318,592 = -313 / 35,975,664 Julian years

In order that the difference between the Hebrew year count and the Julian year count **dHY** becomes 1 Hebrew year, it is necessary that

This result is consistent with the fact thatdHY = 35,975,351 / 313 = 114,937 + 70 / 313 Hebrew years have elapsedor the count in Julisan yearsdJYdJY = 35,975,664 / 313 = 114,938 + 70 / 313 Julian years have elapsed

Correspondents **Larry Padwa and Ari Meir Brodsky** sent a number of delightful observations related to **Hebrew Year 5765H**.

Can you mention at least 3 other calendar related features of Hebrew year 5765H?- began onHebrew Year 5765HThursday 16 September 2004g. - is383 dayslong. - is the8th yearof the304th. - hasmahzor qatanofmolad of Tishrei3d 19h 287p. - was postponed todue toThursday. - is not affected byDehiyyah Molad Zaqenbecause it is a leap year.Dehiyyah GaTaRaD

Correspondents **Larry Padwa and Ari Meir Brodsky** sent a number of other delightful observations related to **Hebrew Year 5765H**.

Here are some:-

Correspondent **Larry Padwa** noted that **Year 5765H** has the ** second** least frequently occurring

The ** molad of Heshvan 5765H** will occur on

The ** molad of Kislev 5765H** will occur on

Correspondents **Larry Padwa and Ari Meir Brodsky** both noted that when a **383-day year** starts on ** Thursday** then there are no double portions in the regular weekly

Correspondent **Ari Meir Brodsky** also found out that the Hebrew and Gregorian calendars for year **5765H** will map onto the exact same dates **152 years** from now. He therefore urges that we keep this year's calendars because these will be reusable **152 years** from now!

Correspondent **Ari Meir Brodsky** drew attention to his fascinating article on the Hebrew year **5765H** How Is This Year Different From All Other Years?

** Yasher Koach!!** to correspondents

[Relative to 5765H] When next will there be a

moladwith a whole number of hours?

As indicated in The Moladot,

It is interesting to note that from month to month theascend by exactlyhalaqimoneunit, going from0 to 17in perfect cyclical order.

29d 12h 793pequal765,433 parts. That is the number of parts which are added to the time of themoladfrom month to month.When

765,433is divided by18there is a remainder of1, thus resulting in the1hldifference from month to month.

Consequently, it will take **1,080 Hebrew months** for all of the excess ** halaqim** to total exactly

**87 years** from now, the ** molad of Tishrei 5852H(2091g)** will be

Correspondents **Larry Padwa and Dwight Blevins** all provided correct answers.

Correspondent **Larry Padwa** solved this problem as follows:

Ignoring whole hours, the increment in the molad from one month to the next is 793 chalakim. There are 1080 chalakim in an hour, and since 793 and 1080 have no common factor, it is necessary to move forward 1080 months to reach the same point in the cycle of chalakim as the starting point. Now to calculate 1080 months from Cheshvan 5765. 1080=4*235 + 140. Thus 1080 months comprise 4 complete 19-year cycles, plus a remainder of 140 months. The complete cycles give 76 years, and the 140 months beginning in year 8 of the machzor katan yield (by brute force calculation) 11 years and 4 months, allowing for leap years at correct intervals. This takes us 76 years + 11 years + 4 months from Cheshvan 5765, which is Adar I 5852. (Whew).

Correspondent **Dwight Blevins** chose this route:

It seems that a forward movement of 1080 months would be required (advancing 1 part per month) for the next molad to fall precisely on the whole hour. Based on that assumption, 1080 months advanced from October 14, 2004, would be 1 Adar 1, February 9, 2092 (5852H). Stated another way, the fall of 2091g, Tishrei 1 falls at 17h 355p. Therefore, 5 more months beyond that date are required to achieve 1080 months from Heshvan, 2004g. Thus, 793 x 5 = 3965p + 355p = 4320p/1080 = precisely 4 hours of parts, past the reference point of Heshvan, Oct. 14, 2004, thus achieving the next whole hour molad, with reference to Heshvan, 5765H. That whole hour molad, again, is 1 Adar 1, 5852H (Feb. 9, 2092).

Thank you Correspondent **Larry Padwa and Dwight Blevins** for providing these very good and very correct answers!

[Relative to 5765H] When next will the

molad of Tishreibe a whole number of hours?

Although, from month to month, the ** halaqim**
ascend by exactly

As explained in Cycles and Moladot,
all periods of Hebrew years that are exact multiples of **19 years**
have exactly one number of months, namely, the number of **19 year**
periods multiplied by **235**.

All other periods of Hebrew years have **two numbers of months**
differing from each other by exactly **one month**.

The ** lower number of months**, for some period of

Consequently, the determination of some formula which will help establish the ** Tishrei moladot** that have

Using empirical computer methods, it is easy to determine that the next ** molad of Tishrei** to have a whole number of hours will be the

What is particularly enticing in this phenomenon is that all of the **whole hour Tishrei moladot** occur in an exact cyclical pattern.

The whole hour

Tishrei moladotoccur in what cyclical pattern?

Although, from month to month, the ** halaqim**
ascend by exactly

As explained in Cycles and Moladot,
all periods of Hebrew years that are exact multiples of **19 years**
have exactly one number of months, namely, the number of **19 year**
periods multiplied by **235**.

All other periods of Hebrew years have **two numbers of months**
differing from each other by exactly **one month**.

The ** lower number of months**, for some period of

Consequently, the determination of some formula which will help establish the ** Tishrei moladot** that have

Using empirical computer methods, it is easy to determine that the next ** molad of Tishrei** to have a whole number of hours will be the

What is particularly enticing in this phenomenon is that all of the **whole hour Tishrei moladot** occur in an exact cyclical pattern.

The pattern can easily be determined by examining a small number of consecutive Hebrew years for Tishrei moladot that contain only whole hours.

These Hebrew years begin with **2H, 264H, 1923H, 2185H, 4106H, 4368H, 6027H, 6289H, 8210H, ...**

From this rather small sample it appears that the indicated cycle of **whole hour Tishrei moladot** is

A bit more research indicates this to be a valid pattern for the entire **Hebrew calendar cycle of 689,472 years**.

The whole day

Tishrei moladotoccur in what cyclical pattern?

Although, from month to month, the ** halaqim**
ascend by exactly

As explained in Cycles and Moladot,
all periods of Hebrew years that are exact multiples of **19 years**
have exactly one number of months, namely, the number of **19 year**
periods multiplied by **235**.

All other periods of Hebrew years have **two numbers of months**
differing from each other by exactly **one month**.

The ** lower number of months**, for some period of

Consequently, the determination of some formula which will help establish the ** Tishrei moladot** that have

Using empirical computer methods, it is easy to determine that the **first molad of Tishrei** to have a whole number of hours will be the

What is particularly enticing in this phenomenon is that all of the **whole day Tishrei moladot** occur in an exact cyclical pattern.

The pattern can easily be determined by examining a small number of consecutive Hebrew years for ** Tishrei moladot** that contain only

These Hebrew years begin with **51171H, 57458H, 63745H, 70032H, 149667H, 155954H, 162241H, 168528H, 248163H, ..., 661008**

From this rather small sample it appears that the indicated cycle of **whole day Tishrei moladot** is

A bit more research indicates this to be a valid pattern for the entire **Hebrew calendar cycle of 689,472 years**.

<--->

How are the significant days of

Hanukkah, that is, the first day ofHanukkahandRosh Hodesh Tevet, related toPesach?

Referring to both The First Day of The Month and The Roshei Hadashim, it is possble to note that according to the fixed Hebrew calendar method

However,the first day ofRosh Hodesh Tevetcan never be onSundaythe first day ofHanukkahcan never be onTuesdaythe first day ofTevetcan never be onThursdaythe first day ofTevetcan never be onShabbat

It is this surprising set of facts derived from the modern fixed Hebrew calendar which give rise to this startling relationship between the significant days of ** Hanukkah** and the festival of

Correspondent **Larry Padwa** opened a topic of such interest that it will lead to the next few ** Weekly Question**s.

Hi Remy, The archives of the Weekly Questions are extremely interesting. On occasion, reviewing old questions can suggest related new ones, as in the following instance. In questions 143 and 145, you and your correspondents discuss the questions of the smallest span of years containing all six possible year lengths (6), and the largest span of years not containing all six possible year lengths (43). A pair of related questions is to identify the smallest span of years containing all fourteen Keviot, and the largest span not containing all fourteen Keviot.

Thank you **Larry Padwa** for this particularly interesting suggestion.

Referring to The *Qeviyyot*, a ** qeviyyah (pl. qeviyyot)** is defined to be a pair of numbers indicating the length of the Hebrew year and the day of the week on which started the Hebrew year of that length.

What is the smallest possible span of Hebrew years to contain all 14 of the

qeviyyot?

Referring to The *Qeviyyot*, a ** qeviyyah (pl. qeviyyot)** is defined to be a pair of numbers indicating the length of the Hebrew year and the day of the week on which started the Hebrew year of that length.

Correspondent **Larry Padwa** opened a topic of such interest that it will lead to the next few ** Weekly Question**s.

**Larry Padwa** correctly assumed the following.

As usual, correspondentHi Remy, It couldn't be less than, since: a)seventeenof theSevenfourteenkeviotare leap years. b) it is impossible to squeeze seven leap years into a sequence smaller than seventeen. To see this, consider a sequence beginning at (say) year 3 of a Machzor Katan. Then seven leap years take us to year 19, yielding a sequence of seventeen years. We cannot do any better than beginning at year 3 (or any other leap year that follows it preceeding leap by three years). If we begin at any other leap year (say year 8 or year 19), then we will require a sequence of 18 years to get seven leap years. And beginning at a common year will also require at least 18. So seventeen is the shortest possible. But that in itself does not prove that a sequence of seventeen years with all fourteen keviot is in fact attainable. -Larry

Correspondent **Robert E. Heyman** sent this very correct calculation result.

Thank you very much correspondentsAccording to my calculations, the smallest possible year span is 18, and the first such span began in the year 888. Robert

The shortest spans of Hebrew years containing all of the 14

qeviyyotbegin in which year, or years, of themahzor qatan(19-year Hebrew calendar cycle)?

Referring to The *Qeviyyot*, a ** qeviyyah (pl. qeviyyot)** is defined to be a pair of numbers indicating the length of the Hebrew year and the day of the week on which started the Hebrew year of that length.

Correspondent **Robert E. Heyman** sent this very correct calculation result.

The year shown by correspondentAccording to my calculations, the smallest possible year span is 18, and the first such span began in the year 888. Robert

Very intriguingly, and by one of these delightful Hebrew calendar coincidences, all of the shortest spans of Hebrew years containing all of the **14 qeviyyot** begin uniquely in the

Perhaps one day, a mathematically inclined Hebrew calendar expert will find the reason behind this mysterious phenomenon.

How many different sequences of

qeviyyotcan be found for the shortest spans of Hebrew years containing all of the 14qeviyyot?

First Begun 21 Jun 1998 First Paged 2 Feb 2005 Next Revised 2 Feb 2005