Notable Properties of Specific Numbers

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Magic Squares

15 is the magic constant for a 3×3 magic square, the smallest possible nontrivial magic square. An N×N magic square consists of the N² integers from 1 to N arranged in an N×N square grid, such that the sum of any row, or any column, or of one of the two diagonals, is equal to any other. There is only one solution for a 3×3 square, not counting its rotations and reflections:

4	9	2
3	5	7
8	1	6

This magic square was known to the Chinese at least 3000 years ago and is called the Lo Shu. It appears in legends and artwork; for an example see here.

As the magic constant of a magic square, 15 is a member of the sequence A6003 (my MCS26973): 1, 5, 15, 34, 65, 111, 175, 260, 369, .... For more about magic squares see the entries for 34, 38, 59, 65, 153, 176, 177, 199, 1514, 1665, 1035369, 1172421, and 276951438.

Triangular Numbers

15 is the sum of the first 5 numbers: 1+2+3+4+5=15. Such numbers are called triangular because 15 things can be arranged in a triangular shape, by putting 1 object in a top "row", 2 in the 2^nd row, and so on down to 5 in the 5^th row. In the OEIS, such numbers are sequence A0217: 1, 3, 6, 10, 15, 21, 28, 35, 45, ... The general formula for the N^th triangular number is N(N+1)/2. Because either N or N+1 is even, we know that either N/2 or (N+1)/2 is an integer. It follows that every triangular number is the product of two integers — either N × (N+1)/2 or N/2 × (N+1) — and therefore, every triangular number above 3 is composite.

The sum 1+2+3+4+5=15 is a little more significant because the first and last numbers, 1 and 5, are also the digits of the total. See 27.

15.154262241479... = e^e

On the x-y plane, the point where the line x=y crosses the curve connecting the nontrivial solutions of x^y=y^x is at x=y=e. At that point x^y equals e^e, and the slope of the x^y=y^x curve is -1.

15.438887358552... = (27/8)^9/4 = (9/4)^27/8

This is the value of x^y for a non-trivial solution of x^y=y^x where x and y are both rational. First, the general solution: Define L to be the log base x of y:

L = log_xy

then x^y=y^x becomes:

x^{x^L} = (x^L)^x = x^{L x}

which (for x>1) reduces to:

x^L = L x
x^L-1 = L

Solving for x, and substituting back for y, we get formulas for x and y in terms of L:

x = L^(1/(L-1))
y = x^L = (L^(1/(L-1)))^L = L^(L/(L-1))

This is the general solution for x^y=y^x with x not equal to y — use any value of L except L=1 and the formulas will give you values for x and y.

We can make x and y both rational by making sure 1/(L-1) is an integer. That is true whenever L is of the form L=(n+1)/n where n is any nonzero integer. Then

x = ((n+1)/n)ⁿ
y = x^(n+1)/n = ((n+1)/n)⁽ⁿ⁺¹⁾

Here are the first few rational (x,y) along with the value of x^y:

n	L	x	y	x^y=y^x
1	2	2	4	16
2	3/2	(3/2)² = 9/4	(3/2)³ = 27/8	(27/8)^9/4 =(3/2)^(3×9/4) =(3/2)^27/4 =(3/2)^(2×27/8) =(9/4)^27/8 =15.438887358552...
3	4/3	(4/3)³ = 64/27	(4/3)⁴ = 256/81	(256/81)^64/27 =15.296931343617...
. . .
infinite	1	e	e	e^e

As n gets bigger, L gets closer to 1 and x and y get closer and closer to each other. For example, when n is 100, x=2.704813... and y=2.731861.... Both converge on e, because

e = lim (1 + 1/n)ⁿ

16 = 2⁴ = 2^2² = 2^④3 = (2²)² = 2_④3, where ^④ and _④ are the higher and lower hyper4 operators.

The only non-trivial solution of x^y=y^x for integer x and y. (This was proved by Euler, and this entry describes a proof.)

hexadecimal numbers

Base 16, commonly called hexadecimal, is the most popular base among computer programmers for representing raw data in computer memory. The "digits" A, B, C, D, E and F are used to represent values of 10 through 15. So, for example, the hexadecimal number 6A₁₆ is 6×16+10 = 106.

Until the mid-1970's, base 8 (called Octal) was the most common base in computer programming applications. The primary reason base 16 overtook base 8 is that it uses 4 bits per digit and 4 itself is a power of 2. All the popular microprocessors, from the very early ones in the 1970's, have been based on a power of 2 bits per machine "word" (size of a register, etc.). Thus, the machine word can be expressed as hexadecimal digits with no bits left over, and the same digits are used for the values you get when splitting the word into top and bottom half-words; whereas octal does not have these benefits.

17 is the only prime number that is the sum of four consecutive primes (2 + 3 + 5 + 7).

17 is a Fermat prime (a prime of the form 2^{2^N}+1) and it is also the exponent of a Mersenne prime (a prime p for which 2^p-1 is prime). The corresponding perfect number is 8589869056.

There are 17 planar crystallographic groups.

Peter, a film critic, enjoys going to eclectic cocktail parties to meet the directors and producers. He has noticed that for any two people in a party, they have always either A) met each other at the studio, B) met each other at the Academy Awards, or C) met each other here because they haven't met before the party. Peter got to wondering, how many people would need to be at a party to guarantee that there is at least one group of three people who have met each other in the same place (A B or C)? The answer is 17.

17 is the smallest number that can be written as A² + B³ in two different ways: 17 = 3² + 2³ = 4² + 1³. By the way, the pair (8, 9) is the only pair of consecutive numbers where one is a square and the other is a cube (Euler proved this.) Catalan's conjecture (proven in 2002) states that 2³ and 3² is the only pair of consecutive powers, aside from trivial cases where one of the numbers is a "1^st power".

The reciprocal of 17, 1/17=0.05882352941176470588235..., has a 16-digit repeating decimal, which is the longest possible. This is a property that 17 shares with 7 and with many higher prime numbers. It results from the fact that, when you perform long division to compute 1/17, every remainder except 0 comes up exactly once. Also because of this, the multiples 2/17, 3/17, and so on have decimal fractions that use the same set of 16 digits, but starting in a different place (just as seen with 1/7, 2/7, 3/7 etc.). A reader named Jeremy pointed out to me that these 16 digits reduce to 9 using the casting out 9's technique, and thus 0588235294117647 is a multiple of 9. This holds for all reciprocals of primes (starting with 7) and is a consequence of Fermat's little theorem.

cult numbers

and psychologically random numbers

These are two closely-related types of numbers. 17 belongs to both classes.

A "cult" number has a "following", a group of "fans", many of whom have set up web pages for the number, a sort of virtual shrine to the number. Typically, a cult-number fan is someone who has one favourite number, and who delights in noticing that number, whenever it occurs in a place that seems to be more than just coincidence. I seem to have chosen 27, because I notice that number a lot more than I "should".

Other cult numbers include 23, 37, 42, 47, 69, and 666; closely related to these are numbers that attract an abnormal amount of quasi-scientific interest, such as the fine structure constant.

A "psychologically random" number, is one that "sounds random", or is chosen more often when someone is asked to pick a random number. 17 is the most often picked number in response to the request "Pick a random number from 1 to 20." Psychologically random numbers are used by writers when some large number is needed but no particular value is better than any other. For example, see 37 for some examples from movies.

Psychologically random numbers are usually 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". This means that most psychologically random numbers are prime. An extended argument along the same lines (attributed to Hilary Putnam by Mark Kalderon), adds that 7, 11 and 13 are nonrandom because of their lucky and unlucky associations, 9 is a square and 3 is "for the Trinity", leaving 17 as the only number less than 20 that isn't special.

When a number is not consciously chosen but just happens by accident, it is more likely to be noticed and perceived as "more than just a coincidence" if the number is psychologically random. When such numbers are noticed repeatedly, they can then become cult numbers. This is why many cult numbers are also psychologically-random numbers.

Here are some links to "cult number" websites:

17: A, B, C

23: A

27: A (formerly here), B, C

37: A

Seventeen (XVII) is considered unlucky in Italy, perhaps because of the resemblance of the Roman numeral to Latin vixi, a euphemism for death. In this way the phobia resembles east-Asian tetraphobia.

(the saros cycle)

18 years is the amount of time it takes for the tilt of the Moon's orbit to rotate a full-circle. Another way of saying the same thing is that the eclipse season slips back an entire year — so every 18 years (more precisely, 18 years 11 days), there have been 38=2×19 eclipse seasons.

Viewed simply as a close match between a multiple of years (18) and eclipse seasons (38), it isn't such a big deal. It's 11 days off, and if you wait a year, you actually get a closer match (40 eclipse seasons is 8 days shorter than 19 tropical years). What makes the 18-year period so special is that the number of synodic months involved (223) is only 0.04 days short of 242 draconic months and only 0.2 days short of 239 anomalistic months. This means that the eclipses, in addition to being at the same time of year, also have the same positioning of Moon and Earth in the other two dimensions (distance and north-south positioning). On each eclipse, the Moon is the same distance away from Earth, and in the same position north-to-south, as it way 18 years 11 days previously. The distance governs whether solar eclipses are annular or total, and the north-south positioning determines where the shadow crosses for both types of eclipses (lunar and solar).

This is the saros, an incredibly rare coincidence that makes eclipses easy to predict. It was discovered by the Babylonians and the knowledge was passed on to the Greeks and thence to later civilisations.

The coincidence is not exact; each saros the Moon is a little further north (on a descending node) or south (on an ascending node) than the previous time, by a distance about equal to 1/70 of the Earth's diameter. So, each repeated eclipse repeats about 70 times, or about 70×18=1260 years.

Also, the saros period is not an exact number of days; the Earth has turned about 1/3 of the way around, so the eclipses (particularly solar) are not seen by the same people.

See also 161178.

To test a number (example 660364) for divisibility by 19:

Remove the last digit, then add 2 times that digit to the remaining number: 66036+2×4 = 66044. (Each time you are adding 19d=(20-1)d and dividing by 10, where d is that removed digit.)
Repeat until it gets down to 20 or less: 6604+2×4 = 6612; 661+2×2 = 665; 66+2×5 = 76; 7+2×6 = 19.
If this process results in 0 or 19 the original number is a multiple of 19; otherwise it isn't.

The Metonic Cycle

19 is the number of years in the repeating pattern of a lunisolar calendar (such as the Hebrew calendar) designed to use months that stay in phase with the moon while also having the year stay in phase with the seasons. 19 tropical years is 6939.60160373 mean solar days, only 2 hours 5 minutes shorter than 235 synodic months (which is 6939.68838046 mean solar days). Due to this rather happy coincidence, the phases of the moon fall on the same dates every 19 years. It takes over 200 years for the error to get to be more than a day (however, in order to accomplish this accuracy, the lengths of at least one of the 235 months must differ from one 19-year cycle to another, see 6940). This cycle was known to several cultures at least as far back as the 4^th century BCE. The period of 19 years also figures in the calculation of the date of Easter (but this is complicated by an additional multiple of 7 due to the fact that Easter must fall on a Sunday, see 133).

Due to another amazing coincidence, 19 tropical years is also (within less than a day) equal to 255 draconic months, which means that eclipses also repeat every 19 years. (However, the pattern only repeats 4 or 5 times before an eclipse stops happening on a given date; the saros is a much better match)

19 is the numerical value of the Arabic word wahid ("one"), one of the names of God. 19 is considered sacred by the Baha'is, an Islamic sect, and they divided the year up into 19 "months" of 19 days each (this makes 19²=361 days; to round out the year an intercalary period of 4 or 5 days is added)⁹. They also group years into 19-year and 361-year cycles (see Bahai calendar).

Someone wrote to me pointing out that 19 has a "special" property: It is the sum of 9 and 10, and also the difference of their squares: 100 - 81 = 19. Of course, this property is true for any number that can be expressed in the form X + X + 1, which means pretty nearly any number (to prove, expand (X+1)² and then subtract X².). This is an example of the type of property that is often reported for cult numbers like 23 and psychologically random numbers like 37, because it is often desirable to find as many properties of such numbers as possible.

19.999099979189475767...

e^π-π. See 23.140692... and xkcd 217 for details.

Joerg Arndt came up with this pretty expression relating this number to 20:

20 = e^π - π + 1/(1111 + 1/(11 + 1/√2)) - ε

where the "error" ε is about 1.21×10^-12 (and noting that in handwriting a pair of 1's can be made to look a bit like π).