Bar-Ilan University
Parashat Hashavua Study Center
Parashat Be-Midbar 5771/ May 28, 2011
Lectures on the weekly Torah reading by the faculty of Bar-Ilan University in Ramat Gan,
Israel. A project of the Faculty of Jewish Studies, Paul and Helene Shulman Basic Jewish
Studies Center, and the Office of the Campus Rabbi. Published on the Internet under
the sponsorship of Bar-Ilan University's International Center for Jewish Identity.
Prepared for Internet Publication by the Computer Center Staff at Bar-Ilan University.
Inquiries
and
comments
to:
Dr.
Isaac
Gottlieb,
Department
of
Bible,
gottlii@mail.biu.ac.il
Prof. Ido Kantor
Department of Physics
The Book of Numbers and Forty Prime Numbers
Within the Pentateuch there are hundreds of words which designate numbers, used to
denote dates, ages, number of sacrifices, number of people in a tribe or group,
measurements of sacred implements, counting actions, elements of the Decalogue, etc. Of
all the books of the Pentateuch, Be-Midbar (Numbers) contains the widest variety of
numbers. Below we shall address the following two questions, in order:
1) How many prime numbers make up the factors of all the numbers that appear in the
Torah? For it is known that any number can be written as a product of prime
numbers (For example, 15 can be factored into 3 x 5, both 3 and 5 being prime
numbers).1
2) What is the order of appearance of the prime numbers comprising all the hundreds
of numbers that appear in the Torah, and how are they dispersed among the five
books of the Pentateuch?
1
A prime number is defined as a number which is divisible only by 1 and by itself. Prime numbers are
important in mathematics and in code theory and were studied extensively by the ancient Greeks in
the third century B.C.E. There is even some evidence that prime numbers were studied by the
Babylonians between 1900 and 1600 B.C.E., and perhaps were even known to the ancient Egyptians.
http://en.wikipedia.org/wiki/History_of_mathematics
1
The answer to the first question is that all the numbers appearing in the Torah can be
factored completely by a set of 40 prime numbers (including the number 1):
1) 24 of these primes appear for the first time in Genesis, 2 new primes are added in
Exodus, 14 new ones in Numbers, and no new ones in Leviticus or Deuteronomy (a
list of the relevant verses is given in the footnotes). This highlights the uniqueness
of the fourth book of the Pentateuch, appropriately called Numbers in English,
which not only contains many numbers but also completes the list of prime numbers
into which all the numbers in the Torah can be factored completely, while Leviticus,
the book that precedes it, as well as Deuteronomy, the book that follows it, contain
no new prime factors.
2) The following prime factors appear in Genesis:2 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 73, 83, 127, 137, 163, 179, 181, 269, 601.
The new prime factors that appear in Exodus are:3 71, 12071.
The new prime factors that appear in Numbers are:4 47, 59, 61, 67, 89, 197, 227,
233, 263, 593, 643, 3167, 4373, 22273.
Now let us address the second question, looking briefly at the order in which the prime
factors appear in the Torah. In the description of Creation in Genesis, chapter 1, the prime
numbers appear in order: 1, 2, 3, 5, 7. After the account of Creation the prime numbers
continue to appear in order – 11, 13: Genesis 4:24, "If Cain is avenged sevenfold, then
Lamech seventy-sevenfold" (11 x 7 = 77); Genesis 5:3, "When Adam had lived 130 years" (13
x 10 = 130). Thus far, the prime numbers have been appearing sequentially. Further on, the
2
Aside from the prime numbers mentioned in this article, the following numbers appear (the prime
factor of the number in the verse appears here in bold face and parentheses): 5:5: "All the days that
Adam lived came to 930 years; then he died" (31); 5:8: All the days of Seth came to 912 years; then
he died" (19); 5:16: "After the birth of Jared, Mahalel lived 830 years and begot sons and daughters"
(83); 5:17: " All the days of Mahalel came to 895 years" (179); 5:25: " All the days of Enoch came to
365 years" (73); 5:25-26: "When Methuselah had lived 187 years … After the birth of Lamech,
Methuselah lived 782 years" (17, 23); 5:31: " All the days of Lamech came to 777 years" (37); 8:13:
"In the six hundred and first year" (601); 11:17: "After the birth of Peleg, Eber lived 430 years" (43);
11:22: "When Nahor had lived 29 years" (29); 11:32: "The days of Terah came to 205 years" (41);
23:1: "Sarah's lifetime – the span of Sarah's life – came to one hundred and twenty-seven years"
(127); 25:17: "These were the years of Ishmael: one hundred and thirty-seven years" (137).
3
38:25: "The silver of those of the community who were recorded came to 100 talents and 1,775
shekels by the sanctuary weight" (71); 38:26: "for each one who was entered in the records, from the
age of twenty years up, 603,550 men" (12,071).
4
Aside from the numbers mentioned explicitly in this article, 1:23: "those enrolled from the tribe of
Simeon: 59,300" (593); 1:37: "those enrolled from the tribe of Benjamin: 35,400" (59); 1:43: "those
enrolled from the tribe of Naphtali: 53,400" (89); 2:9: "The total enrolled in the division of Judah:
186,400" (233); 2:24: "The total enrolled in the division of Ephraim: 108,100" (47); 2:31: " The total
enrolled in the division of Dan: 157,600" (197); 3:43: "All the first-born males as listed by name,
recorded from the age on one month up, came to 22,273" (22,273); 4:42: "those recorded by the
clans of their ancestral house came to 2,630" (263); 26:50: "those are the clans of the Naphtalites,
clan by clan; persons enrolled: 45,400" (227); 26:51: "This is the enrollment of the Israelites:
601,730" (3,167); 31:39: "the asses came to 30,500" (61); 31:52: "All the gold that was offered by
the officers of thousands and the officers of hundreds as a contribution to the Lord came to 16,750
shekels" (67).
2
numbers begin to appear in disarray, with considerable jumps and not in ascending order.
Five verses after the appearance of the number that can be factored by 13, the next prime
factor which appears is 269: Genesis 5:7, "After the birth of Enosh, Seth lived 807 years"
(260 x 3 = 807).5 Could it be that as evil and corruption became rooted and spread in the
world, after the order that there was in Creation and prior to the flood, also the prime
numbers that are the foundation of modern code theory suddenly began to appear in
disarray and not in sequence?
Further investigation of the appearance of the prime factors indicates that in the first census
of the Israelites, in Parashat be-Midbar, and in the second census, in Parashat Phinehas, not
only are there differences in the total count of the Israelites, but also the different numbers
for the tribes lead to the birth of twelve new prime factors (including the figure for the firstborns), among them some of the largest prime factors in the Torah and numbers that appear
in the Torah only once. For example, the second census of the tribe of Issachar (Numbers
26:25), "Those are the clans of Issachar; persons enrolled: 64,300" (643), and the second
census of the tribe of Reuben (Numbers 26:7): "Those are the clans of the Reubenites. The
persons enrolled came to 43,730" (4373 x 10 = 43,730).6
We conclude by noting that the first 21 prime numbers in the sequence of numbers (see the
chart) appear in our above-mentioned list of 40 prime numbers, save for the number 53,
which does not appear in the Torah.
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73
Perhaps the Torah is hinting to us that Gan (53 in gematria) Eden, the garden of Eden, is
closed off to human beings, as the Torah notes in Genesis 3:23-24: "So the Lord G-d
banished him from the garden of Eden, to till the soil… He drove the man out, and stationed
east of the garden of Eden the cherubim and the fiery ever-turning sword, to guard the way
to the tree of life." And perhaps the absence of this number alludes to the gene (Hebrew
gen, also 53) that was subtracted from man after he sinned, as we read in Genesis 6:3: "My
breath shall not abide in man forever, since he too is flesh; let the days allowed him be one
hundred and twenty years."
We noted in our introductory words that the verses of the Torah contain hundreds of
numbers, and we have shown that all the numbers in the Torah can be factored completely
by no more than 40 prime numbers and that these prime numbers are concentrated in the
books of Genesis and Numbers. Closely examining the Torah, one finds that all 54 weekly
5
269 is bases on the prime number (59). The prime number that follows 13 is 17, and the age 816
(which is close to 807), is divisible by 17, indicating an inclination towards continuing the prime
numbers in order.
6
The last prime factor, 67, appears at the end of chapter 31 of Numbers: "All the gold that was
offered by the officers of thousands and the officers of hundreds as a contribution to the Lord came
to 16,750 shekels" (67). It is interesting that at the beginning of this chapter Moses was told, "Avenge
the Israelite people on the Midianites; then you shall be gathered to your kin" (31:2); and the process
of giving the Torah to the people of Israel, for which Moses had to spend 40 days and 40 nights on
Mount Sinai, was not complete until all its components had been given.
3
readings of the Torah contain numbers (not necessarily prime ones), save for only one
reading: Parashat Nitzavim (the 51st weekly portion), which is comprised of 40 verses.
4