# Notable Properties of Specific Numbers

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The sum of the rows, columns, and diagonals of a 4×4 perfect square. See 1514 for an example.

In general, for an N×N magic square, the sum of a
row, column or diagonal is N(N^{2}+1)/2 or (N+N^{3})/2. That sequence
runs: 1, 5, 15, 34, 65, 111, 175, ... (although there are no 2×2
magic squares). This is Sloane's sequence A6003.

One of the numbers that appears down the middle of Pascal's triangle (see that entry for more). See also 252.

35 is also a semiprime.

36 is the smallest number that is both square and triangular.

36 = 5 + 7 + 11 + 13 = 17 + 19. The smallest number that can be expressed as the sum of consecutive primes in two different ways. It is also the smallest square with the property that N/2-1 and N/2+1 are both prime.

36 is often used as a base for positional numerals, typically because one can use the ten digits plus the 26 letters of the Latin alphabet as the 36 needed symbols. The values of the letters are similar to hexadecimal, extended by G=16, H=17, etc. all the way to Z=35.

37 is a hexagonal number: 37 = 1 + 6 + 12 + 18.

37! + 1 is prime.

1/37 = .027027027..., and 1/27 = .037037037... This is related to the fact that 37×27=999. If you replace both numbers with their average, you get 32×32=1024, and 1024 is another number with important properties related to the fact that it is close to 1000.

The divisibility of 999 and 111 by 37 gives us the following kind-of-easy test for divisibility by 37:

- Take the digits of the number in groups of 3 starting from the right, and add the resulting numbers together. If the result is more than 3 digits, repeat this process.
- Subtract 111 as many times as you can without going below 0.
- If the result is 0, 37 or 74, the original number is divisible by 37; otherwise it isn't.

The average healthy human body temperature is 37^{o} Celsius.

37 is a psychologically random number, similar to 17 and 27 and having no particular cultural origin. As is mentioned under 17, some numbers are often picked by people when they need a number that sounds "random". Such numbers are odd and don't end in 5, because there is a natural psychological bias to thinking even numbers and numbers that end in 5 are "less random". As a result, numbers that end in 1, 3, 7, and 9 occur more often than they "ought to". Here are some examples of 37's in movies:

- In Monty Python and the Holy Grail, King Arthur (played by Graham Chapman) encounters a peasant and calls him "old woman"; the peasant (Michael Palin) replies that he's a man and furthermore "I'm 37! I'm not old!"
- In Phenomenon, the main character (played by Travolta) begins experiencing the phenomenon on his 37th birthday.
- In Casablanca, Rick (played by Humphery Bogart) is 37 years old.
- In Pulp Fiction, right after a line "I'll be there in ten" we see a title card saying "Nine minutes and thirty-seven seconds later".
- Many movies use 37's for such things as taxi cab numbers, prison cell numbers, addresses, etc.
- Movie titles: Horror at 37,000 Feet (1972, TV), Fire! Trapped on the 37th Floor (1991, TV), To Gillian on her 37th Birthday (1996), 37 Stories About Leaving Home (1997).

There is only one possible magic hexagon (defined similarly to a magic square:

3 17 19 18 7 16 1 2 11 5 12 6 4 9 8 10 14 13 15When you add the 3, 4 or 5 numbers in a row (that is, any row parallel to an edge of the hexagon), the total is 38. this is the only size magic hexagon that is possible.

38 in Roman numerals ("XXXVIII") comes later in alphabetical order than any other number. The first would be 100 ("C").

39 is a "composite" number because it is the product of two or more prime numbers: 3×13=39. Determining whether a number is composite is equivalent to determining if a number is prime, but proving that a number is composite is usually much faster, because there are many "pseudo-prime" tests. A pseudo-prime test is always valid when used as a test for compositeness, but not always valid for tests of primeness.

39 = 3 × 13 = 3 + 5 + 7 + 11 + 13, the sum of a sequence of consecutive primes and the product of the first and last primes in that sequence. The first such sequence is 10 = 2 × 5 = 2 + 3 + 5; after 39 there are at least three more: 155, 371 and 454539357304421. (Related to Sloane's sequences A55233 and A55514).

The smallest nuber with a persistence of 3: 3×9 = 27; 2×7 = 14; 1×4 = 4.

Another type of "persistence" can be defined using alternating
exponents and multiplications: 3^{9} = 19683;
1^{9}×6^{8}×3 = 5038848; 5^{0}×3^{8}×8^{4}×8 = 214990848;
and so on. This continues for another 6 steps and you get 0,
so 39 has a "powertrain persistence" of 9, which is more than any
smaller number. The next number to set a record is 3573.

Unaware of the "sum of sequence of primes" property, David
Wells^{4} (in the first version of his book) called 39 the
"smallest uninteresting positive integer", i.e. the first number with
no particularly special attributes. (In the second edition of his
book, that honor went to 51.) This is usually used as example
of the type of contradictions one encounters when considering
self-referential statements (if it's the first uninteresting number,
then that makes it interesting, right?). I prefer to look at it as an
example of the never-ending depth of study in a field: If you look
hard enough and long enough, you can find something interesting about
any number. In the case of 39, the "sum of primes" property
mentioned above was found only in the last 50 years, which is
relatively recent for such a small number. (Of course, 39 has been
linked to 666, but so has everything else.)

My first uninteresting number is currently here; see also 12407.

An approximation for the number of inches in a meter used in the United States prior to 1975. Its reciprocal 1/39.37 is 0.0254000508001016002032...; see 2.54 and 99.9998.

There are frequent Biblical references to 40 (40 days of rain, 40 years in the wilderness, 40 lashes, 40 days of Lent, etc.) and some of these probably result from 40 being a common idiom for "many".

By contrast, "40 acres and a mule" (the unfulfilled promise to freed African American slaves by General Sherman) probably gets its "40" from the fact that 40 acres is 1/16 of a square mile.

40 ("forty") is the only number whose name in English has all of its letters in (ascending) alphabetical order.

Euler discovered that the polynomial x^2+x+41 gives a prime number for all x from 0 to 39. There was much speculation about whether some polynomial or similar formula could give primes for all values of its variables, and that was eventually proven impossible (Goldbach, 1752).

There are other, similar polynomials with smaller constant terms,
for example x^{2}+x+17. They are related to the same set of
numbers that cause near-integer values of e^{π√n} (see
163). The connection is more obvious if you use the
quadratic formula: from x^2+x+41 we get
(-1{+-)√1-4×41)/2, or (-1{+-)√-163)/2. Alternately the
polynomial can be broken up into a squared monomial plus a constant:
x^{2}+x+41=(x+^{1}/_{2})^{2}+^{163}/_{4}. Amazingly,
this is connected to the Monster group, due to the
Monstrous moonshine phenomena.

Although this Euler polynomial fails to give primes for all x, there are formulas that have multiple variables for which, if the value of the formula is positive, it is prime. There are even ones for which every prime can be found as the value of the formula for some set of positive values of the variables. Such a formula was found by J.P. Jones, Hideo Wada, Daihachiro Sato and Douglas Wiens, and published in 1976:

(k+2){ 1 - [wz+h+j-q]^{2}
- [(gk+2g+k+1)(h+j) + h - z]^{2}
- [2n+p+q+z-e]^{2}
- [16(k+1)^{3}(k+2)(n+1)^{2} + 1 - f^{2}]^{2}
- [e^{3}(e+2)(a+1)^{2} + 1 - o^{2}]^{2}
- [(a^{2}-1)y^{2} + 1 - x^{2}]^{2}
- [16r^{2}y^{4}(a^{2}-1) + 1 - u^{2}]^{2}
- [((a + u^{2}(u^{2}-a))^{2} - 1) (n+4dy)^{2} + 1 - (x+cu)^{2}]^{2}
- [n+l+v-y]^{2} - [(a^{2}-1)l^{2} + 1 - m^{2}]^{2}
- [ai+k+1-l-i]^{2}
- [p + l(a-n-1) + b(2an+2a-n^{2}-2n-2) - m]^{2}
- [q + y(a-p-1) + s(2ap+2a-p^{2}-2p-2) - x]^{2}
- [z + pl(a-p) + t(2ap-p^{2}-1) - pm]^{2} }

The number of variables can be reduced to 10. However, that requires
massively increasing the size of the exponents (possibly as high as
1.6×10^{45} — shown by J.P. Jones in 1982).

The Catalan numbers 1, 1, 2, 5, 14,
42, 132, 429, 1430, 4862, 16796, 58786, ...
(Sloane's A0108) are generated by the formula
C_{n}=(2n)!/(n!×(n+1)!). For example, C_{5} =
10!/(5!×6!) = 3628800/(120×720) = 42. The following illustrates
the first few Catalan numbers (notice the similarities and differences
to the Motzkin numbers):

In the above diagram, notice that the paths can be grouped by how
many "peaks" each has, and the parenthesis strings by how many times
"()" occurs. For example, the C_{4}=14 case is broken up into
1+6+6+1. Each Catalan number can be broken up in this way, and
arranged into a triangle similar to Pascal's triangle for
which each row adds up to a Catalan number:

In general, these numbers are called the Narayana numbers after the
Indian mathematician; Sloane's sequence A1263
gives them. This triangle can be generated from Pascal's triangle by
taking the N^{th} number in each row, multiplying by the number to
its lower-left and dividing by N. For example, the 4^{th} number in
the row 6 of Pascal's triangle is a 10, and the number 20 is to its
lower-left. 20×10/4 = 50, which gives the 4^{th} number in the row 6
of the Catalan triangle.

To generate a row from scratch: start with 1, then multiply N times by a fraction whose numerators and denominators step sequentially forwards and in reverse (respectively) through the first N triangular numbers: 1; 1×21/1 → 21; 21×15/3 → 105; 105×10/6 → 175; 175×6/10 → 105; 105×3/15 → 21; 21×1/21 → 1.

In the second diagonal (1, 3, 6, 10, 15, ...) are the triangular numbers; the third diagonal (1, 6, 20, 50, 105, 196, ...) are a four-dimensional analogue of the tetrahedral and pyramidal numbers (see 196), and the fourth diagonal (1, 10, 50, 175, ...) can be interpreted as a six-dimensional figurate number, which is discussed here.

The Catalan numbers are also related to the Bell numbers.

42 days = 42×24×60×60 = 3628800 seconds, and this is exactly 10! (10 factorial), because 42×24×60×60 = 7×6 × 8×3 × 3×10×2 × 5×4×3 = 7×6 × 8 × 9 × 10×2 × 5×4×3 = 1×2×3×4×5×6×7×8×9×10. See also 10080, 40320, 604800, and 86400000.

"42 months" is mentioned twice in the book of Revelation; see 1260.

42 is a cult number, whose cult status definitely
originated with Douglas Adams' use of 42 as "The answer to the
ultimate question of life, the universe and everything" in his
Hitchhiker's Guide to the Galaxy series. Most current pop
culture references to 42 derive from this. (See also
2.74858523...×10^{80588}).

Lewis Carroll also mentioned 42 a few times in his writings, but did
not manage to make 42 into a cult phenomenon the way Adams has.^{106}

(Gratuitous connections with 27: 3×(2×7)=42 but 3×(2+7)=27; (42 + 4×2 + 4)/2 = 27.)

43 is a prime number, an integer that can not be divided evenly by another integer other than itself or 1.

Testing numbers to see if they are prime becomes (famously) difficult as the number to be tested gets bigger. There are various tests to quickly rule out the possibility of being prime (see composite, but no easy way to rule out all such possibility.

The sequence 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... is generated by taking the product of the first n terms and adding 1 (for example, 2×3×7+1=43). This is called Sylvester's sequence, and is Sloane's sequence A0058. Because of the way the terms are calculated, no two terms in the sequence have a common factor. Because of that, no two terms have a common prime factor, and therefore there are an infinite number of primes. See also 47.

The factorials count the number of different ways n items can be arranged. The subfactorials, also called rencontres numbers, count the number of ways they can be arranged such that none is in its "proper" place. For example, three things (A,B,C) can be arranged 6 ways, but only two (C,A,B) and (B,C,A) have none of the items in their "original" place. The subfactorials are: 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, ... They can be generated by:

A_{1} = 0

A_{n} = n A_{n-1} + -1^{n}

or (curiously enough) by:

A_{n} = round(n!/e)

(the "round" function rounds to the nearest integer). Due to that
formula, if you shuffle n objects randomly, there is about a 1/e
chance that none of them will be in its original position. This in
turn is related to the definition of e — the limit (as n
approaches infinity) of (1+1/n)^{n}.

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Quick index: if you're looking for a specific number, start with whichever of these is closest: 0.065988... 1 1.618033... 3.141592... 4 12 16 21 24 29 39 46 52 64 68 89 107 137.03599... 158 231 256 365 616 714 1024 1729 4181 10080 45360 262144 1969920 73939133 4294967297 5×10

^{11}10

^{18}5.4×10

^{27}10

^{40}5.21...×10

^{78}1.29...×10

^{865}10

^{40000}10

^{9152051}10

^{1036}10

^{1010100}— — footnotes Also, check out my large numbers and integer sequences pages.

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