# Notable Properties of Specific Numbers

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34

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

In general, for an n×n normal magic square, the sum of a row, column or diagonal is n(n2+1)/2 = (n+n3)/2. (The "normal" qualifier means that the numbers in the square run from 1 to n2). That sequence runs: 1, 5, 15, 34, 65, 111, 175, ... (although there are no 2×2 magic squares). This is Sloane's sequence A6003.

-34 ("negAtivE thIrty fOUr") is the smallest (in absolute value) integer to have the vowels A,E,I,O,U in order in its name. It's also the "smallest" such number-name in the sense of having the fewest letters. See also 84, 1005, 1025, 1084, 5000, and 1000000000008020.

35

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

36 is the smallest number that is both square and triangular, and 36's square root is also triangular. See 6 and 204 for more on this.

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

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 37o 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).

38

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 15

When 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").

composite numbers

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 number 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: 39 = 19683; 19×68×3 = 5038848; 50×38×84×8 = 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 Wells4 (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.)

39.37

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.

40

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.

41

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 x2+x+17. They are related to the same set of numbers that cause near-integer values of eπ√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: x2+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 - f2]2 - [e3(e+2)(a+1)2 + 1 - o2]2 - [(a2-1)y2 + 1 - x2]2 - [16r2y4(a2-1) + 1 - u2]2 - [((a + u2(u2-a))2 - 1) (n+4dy)2 + 1 - (x+cu)2]2 - [n+l+v-y]2 - [(a2-1)l2 + 1 - m2]2 - [ai+k+1-l-i]2 - [p + l(a-n-1) + b(2an+2a-n2-2n-2) - m]2 - [q + y(a-p-1) + s(2ap+2a-p2-2p-2) - x]2 - [z + pl(a-p) + t(2ap-p2-1) - 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 1.6×1045 — shown by J.P. Jones in 1982).

41 is the smallest factor of 99999 that does not divide into any smaller 999-number, so it is the smallest integer whose reciprocal has a 5-digit repeating decimal, 1/41 = 0.02439024390243902439... For more on such things, see my separate article Fractions with Special Digit Sequences; see also 89, 99.9998, 998, 9801, 9899, and 997002999.

The Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... (OEIS sequence A0108) are generated by the formula Cn=(2n)!/(n!×(n+1)!). For example, C5 = 10!/(5!×6!) = 3628800/(120×720) = 42. The following illustrates the first few Catalan numbers (notice the similarities and differences to the Motzkin numbers):

Left side illustrates: Right side illustrates: Number of paths of N steps Number of distinct ways N where each step goes pairs of parentheses can be northeast or southeast, and arranged, in matching (possibly never goes further south nested) pairs. than the starting point.     C = 1 /\ ()     /\ C = 2 / \ /\/\ ()() (())     /\ / \ /\/\ /\ C = 5 / \ / \ /\/ \ ((())) (()()) ()(())   /\ / \/\ /\/\/\ (())() ()()()     /\ / \ / \ C = 14 / \ (((())))   /\ /\/\ /\ /\/ \ / \ / \/\ / \ / \ / \ (()(())) ((()())) ((())()) /\ /\ / \ /\ /\ / \ /\/ \ / \/ \ / \/\ ()((())) (())(()) ((()))()   /\ /\ /\ / \/\/\ /\/ \/\ /\/\/ \ (())()() ()(())() ()()(())   /\/\ /\/\/\ /\/\ / \/\ / \ /\/ \ (()())() (()()()) ()(()())   /\/\/\/\ ()()()()

The Narayana 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 C4=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:

1 1 1 1 3 1 1 6 6 1 1 10 20 10 1 1 15 50 50 15 1 1 21 105 175 105 21 1 1 28 196 490 490 196 28 1 1 36 336 1176 1764 1176 336 36 1

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 Nth number in each row, multiplying by the number to its lower-left and dividing by N. For example, the 4th 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 4th 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 as a Cult Number

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...×1080588).

Lewis Carroll also mentioned 42 a few times in his writings, in both obvious and obscure ways. He did not manage to make 42 into a cult phenomenon the way Adams has106; however it is likely that Carroll's usage influenced Adams'. See also 74088.

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

prime numbers

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.

mutually co-prime sequences

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" or "derangements", 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: !1 = 0; !2 = 1; !3 = 2; !4 = 9; !5 = 44; !6 = 265; !7 = 1854; !8 = 14833; !9 = 133496; !10 = 1334961, ... (OEIS sequence A0166). They can be generated by:

A1 = 0
An = n An-1 + -1n

or (curiously enough) by:

An = 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.

Numberphile has a nice video on the topic, Derangements and they explain the derivation of the n!/e approximation in extra footage.

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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    5×1011    1018    5.4×1027    1040    5.21...×1078    1.29...×10865    1040000    109152051    101036    101010100    — —    footnotes    Also, check out my large numbers and integer sequences pages. s.11