| Notable Properties of Specific Numbers |
Back to page 9 . . . Forward to page 11
199 is prime, and if you add 210 over and over again, you
get 8 more primes. These 9 primes together can be arranged into a
3×3 magic square, and is the smallest possible magic square made of
primes in arithmetic progression:
1669 199 1249
619 1039 1459
829 1879 409
See also 177.
199 is also a Woodall number: 2×102-1.
204 = 17*12, the product of the numerator and denominator of 17/12, which is an approximation to the square root of 2 formed by the Pell numbers and another similar sequence. The first few numbers like this are: 1, 6, 35, 204, 1189, 6930, 40391, 235416, 1372105, 7997214, ... (Sloane's integer sequence A001109). The sequence follows the pattern:
An = 6 An-1 - An-2
(for example, 204=6×35-6 and 1189=6×204-35).
A really nifty property of these numbers is that their squares are also triangular — see 41616 for more.
210 is a primorial: the product of the first N primes: 210=2×3×5×7. The primorials are: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, ... (Sloane's A002110).
210 is a 3-d oblong number, the product 5×6×7. Since it is also 14×15, it is another example of a number that is the product of consecutive integers in two different ways (see also 120, 720, 5040).
For a while, 2 is the most common gap between consecutive prime numbers. Then (starting at about 100, and permanently after 1000), gaps of 6 occur most often. Up at around 1035, it shifts to 30. There is a conjecture that above about 10425 the most common prime gap is 210.
Base 210 is also a record-setter for tests for divisibility by primes. See 66.
216 is a cube, 63, and is the sum of three consecutive cubes72: 33 + 43 + 53 = 63. See also 25, 143, 8000 and 31858749840007945920321.
8 pieces are required to construct a puzzle that can be assembled as
a 6×6×6 cube and reassembled into the three smaller cubes. The 6
"slices" of the puzzle would appear as follows, with letters a
through h representing the eight puzzle pieces:
f f f f f h f f f f f h f f b b b b
f f f f f e f f f f f e f f b b b b
f f f f f e f f f f f e f f f b b b
g g g e e e g g g e e e g g g b b b
g g g e e e g g g e e e g g g e e e
g g g e e e g g g e e e g g g e e e
slice-1 slice-2 slice-3
.
f f b b b b a a b b b b a a b b b b
f f b b b b a a b b b b a a b b b b
c c b b b b c c b b b b c c b b b b
c c b b b b c c b b b b c c b b b b
c c c e e e c c c e e e c c c d d d
c c c e e e c c c e e e c c c d d d
slice-4 slice-5 slice-6
216 = 8×27 = 23×33, and had spiritual significance in ancient times because it was the product of the first two cubes (they didn't count 13 as a cube).
It is also 6×6×6, and is the value of Φ(666), the number of integers less than and relatively prime to 666.
In the 1998 movie π (a.k.a. "Pi the Movie"), the Qabbalistic Jews are searching for a sequence of 216 Hebrew letters, or a 216-digit number (which is the same thing in ancient Hebrew, since the letters of the Hebrew alphabet are used to represent the digits of numbers).
220 and 284 form an amicable pair, an idea that goes back to the time of Pythagoras. If you add the factors of 220 you get 284 and vice versa. This is related to the idea of perfect numbers.
231 appears as a denominator in the "greedy" Egyptian fraction for 3/7: 3/7 = 1/3 + 1/11 + 1/231.
To the best of our knowledge, the ancient Egyptians had no general notation for an arbitrary fraction A/B. They had a special symbol for 2/3, and for any other fractions were expressed as a sum of "unit fractions", fractions of unity, such as 1/3 and 1/7. For example, instead of 3/4 they wrote 1/2+1/4, and instead of 5/6 they wrote 1/2+1/3. By convention, they always expressed it as a sum of different unit fractions (for example, 2/7 could not be written 1/7+1/7, but 1/4+1/28 or 1/5+1/20+1/28 were OK), and the fractions were always listed largest-first (that is, smallest denominators first).
It is not entirely known why they did it this way. It could have been for practical reasons in performing physical divisions. For example, imagine dividing 2 equal bushels of grain among 7 people. The simple approach would be to divide each into 7 equal parts. But a fair division into 7 parts is difficult; 4 is much easier (use a balance, or the "I split, you choose" method, to divide in half; then repeat). So, divide each bushel into 4 equal parts, and give 1/4 bushel to each person; the remaining 1/4 bushel then is divided into 7, which is much easier than the original task.
It is possible to express any fraction as a sum of unit fractions, and usually in more than one different way. There are several different algorithms for converting a fraction like 3/7 into an Egyptian fraction sum. One, called the "greedy" algorithm, works by subtracting the largest possible unit fraction each time and repeating:
Given: Fraction X/Y, where X and Y are positive integers and X
2: 1/A is the first (or next) term in the Egyptian fraction.
3: Compute X/Y - 1/A and reduce to lowest terms. The result is
the new X/Y. If nonzero, go back to step 1 and repeat.
This algorithm always produces an answer with a finite number of terms, but the denominators sometimes get really large.
This prime number appears in one of the earliest known geometrically converging formulas for computing π:
π/4 = 4 arctan(1/5) - arctan(1/239) = SUM [ (-2n)/(2n+1)(52n+1) - (-1n)/(2n+1)(2392n+1) ]
This formula works because of this special relationship between 5 and 239 through their squares:
2 × 134 = 2392 + 1 = 57122
and
2 × 13 = 52 + 1 = 26
and
arctan(x) + arctan(y) = arctan( (x + y) / (1 - x y) )
This relationship makes it possible to show that a geometric construction of 4 right triangles in the proportion 1 :: 1/5 :: (√26)/5 and one triangle in the proportion 1 :: 1/239 :: (√57122)/239 can be used to produce a triangle in the proportion 1 :: 1 :: √2. Also related to this is the fact that 239/169 = 239/132 is a good close approximation to the square root of 2.
You might wonder, why do all that when you could compute π/4 directly from the arctangent of 1:
π/4 = arctan(1) = SUM [ (-1n)/(2n+1) ] = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
The reason is that this series converges too slowly to be of any use. The 52n+1 in the denominator in the series above makes it converge quickly enough so that you get about 1.398 = log10(52) digits of π for each term you evaluate.
239 is the smallest factor of 9999999 that is not also a factor of some smaller string of 9's, and therefore 239 is the smallest number whose reciprocal has a 7-digit repeating pattern: 1/239=0.00418410041841004184... See also 27 and 757.
Here are two rather obscure properties of 239:
239 is the largest number that cannot be expressed as the sum of 8 (or fewer) cubes (it requires 9: 239 = 53 + 33 + 33 + 33 + 23 + 23 + 23 + 23 + 13). The only other such number is 23. 239 also requires the maximum number of terms to be expressed as a sum of squares (4 squares, 239 = 152 + 32 + 22 + 12) or of 4th powers (19 4th powers). It does not require the maximum number of 5th powers.
Define the "sum of prime factors" sopf(N) to be the sum of each of the prime factors of N, counting a prime more than once if N is a multiple of its square, cube, etc. So, for example, sopf(42) = 2 + 3 + 7 = 12, and sopf(27) = 3 + 3 + 3 = 9. sopf(N) = N only if N is prime, or if N is 1 or 4. Now consider the sum N + sopf(N). As N increases, this value increases irregularly. Some values never occur — for example, there is no N such that N + sopf(N) equals 12 — and other values occur more often — for example, there are two 14's and two 23's. The value 239 occurs often enough that if you add the sopf(N)'s for all N's that have N+sopf(N)=239, you get a sum greater than 239. 239 is the first number for which this occurs (the next few are 1439, 2159, 4319).
(Personal: 239 is another street number where I have lived. The famous HAKMEM72 file was MIT AI memo number 239.)
240 is a "Fibonacci factorial", the product of the first 6 Fibonacci numbers: 1×1×2×3×5×8 = 240. See also 158.
251 is the first of a set of 4 consecutive primes that are spaced an equal distance apart: 251, 257, 263, and 269 are all prime, there are no primes in between, and the spacing between them is 6. 251 is the lowest number with this property; the next is 1741. See also 47, 9843019, 121174811 and 19252884016114523644357039386451.
252 = 10×9×8×7×6 / (5×4×3×2×1)
If you flip a coin 10 times in a row, there are exactly 252 ways in which it can turn out that you get exactly 5 heads and 5 tails (some examples are HTHTTTTHHH, THTHTHTHTH, HHHHHTTTTT, TTHHHHTHTT). Note the rather elegant way of expressing this using the numbers 1 through 10. This type of expression can be used to calculate any of the numbers down the center of Pascal's triangle. See also 2520.
This is 28, and is the number of possible values that can be stored in a byte, the most common unit of measurement for computer memory quantities. The number of values results from the fact that there are eight bits in a byte, and each bit can have two values (0 or 1). (Yes, during one period of history (16th century and earlier) there were also eight "bits" in a Spanish dollar, a fact that is related to the phrase "pieces of eight", and "two bits" (as in "shave and a haircut — two bits", slang for "25 cents"). The Spanish dollar was a gold coin with a value of eight reale, and was sometimes actually cut into eight wedge-shaped pieces to make change.)
Eight bits became the standard computer memory quantity because it is also a power of 2 (other smaller "bytes" were used in the early days when memory was expensive, but having the number of bits be a power of two makes many operations more efficient.) The next smaller power-of-2 size would be 4 bits per "byte", but that only allows for 16 = 24 possible byte values. The need to store and manipulate text, including a full alphabet (26 characters in the country where most of the computers were being developed) plus digits and many punctuation symbols meant that a byte that takes at least 100 or so values was necessary.
If computers had been developed in one of the many countries that have over 256 writing symbols (those that use the Korean, Ethiopic, Japanese, Chinese, and Vietnamese writing systems, among others) they might well have settled on a 16-bit byte. In fact, the emerging world standard for text (Unicode) uses 16-bit character codes, and allows distinct values for every symbol in every popular writing system.
257 is a Fermat number, a number of the form 22N+1. Fermat hypothesized that all of these were prime, but they are not. A discussion of them and their factors is here. See also 641.
(Personal: the course number of my only Men's Weekend production)
The number of days in the tzol k'in, the almanac cycle Mayans used for divination (a system that originated at least 2600 years ago and is still used by native Americans in Yucatan, Guatemala, Belize, and Honduras). The days in the tzol k'in have successive names from a set of 20 (similar to the 20 days in the Mayan civil week) and numbers from a set of 13. There are thus 13×20 combinations of a name and a number and it takes this many days for the cycle to repeat. It is not known why 260 was "chosen" although there are many theories. It is well-known the Mayans recognized a period of 52×365 days (slightly less than 52 tropical years), which is exactly equal to 73×260 — but this does not line up with other important astronomical periods (the eclipse seasons, the lunar month or the synodic period of Venus). See also 1872000.
See 220.
This is the value of "4 superfactorial" by the lower (Simon and Plouffe 1995) definition of "superfactorial".
Barnes' G-function is to superfactorials as the Gamma function is to normal factorials. Barnes' G-function can be (very slowly) calculated by the formula:
G(z) = 2πz/2 e-[z(z+1)+γ z2]/2 PRODUCT(n=1..inf)[(1+z/n)n e-z+z2/(2n)]
where γ is the Euler-Mascheroni constant. For sufficiently large values of z you can use the approximation:
G(n) ≅ (e1/12/A) nn2/2-1/12(2π)n/2e-3n2/4
where A is the Glaisher-Kinkelin constant. See here for more.
See also 1010102.0691973765×1036305. There is also a hyperfactorial.
I recently moved, and my house number changed from 27 to 304. I wanted to find relationships between these two numbers, and quickly found two rather straightforward ones:
304 = 3 × 10 × 10 + 4 and 27 = 3 + 10 + 10 + 4 (change × to +)
304 = 24 × 19 and 27 = 2 × 4 + 19 (change ^ to × and × to +)
This sort of thing is very common.
323 = 17×19, a product of two primes that, together with 1001 and 2001, create the rather nice digit pattern of 6469693230.
A "3-dimensional oblong number", the volume of a rectangular solid of dimensions 6×7×8. If you add the middle number (7) you get 73=343. Numbers of this type are always a multiple of 6, because in any three consecutive integers there is always exactly one multiple of 3 and at least one even number. The sequence starts: 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, ... (Sloane's A007531).
A "generalized oblong number" is the product of 2 or more consecutive positive integers. These include the regular oblong numbers, 3-way products such as 336, and numbers like 120 formed by the product of 4 or more integers. The sequence starts: 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, ... (Sloane's A045619). Each is the value of nPr for some values of n and r, n>r and r>1.
For all prime numbers p (except 2), 2p-1 modulo p is 1:
3 is prime, 23-1 = 4 = 3×1+1
5 is prime, 25-1 = 16 = 5×3+1
7 is prime, 27-1 = 64 = 7×9+1
11 is prime, 211-1 = 1024 = 11×93+1
13 is prime, 213-1 = 4096 = 13×315+1
etc.
This is also true for some (but only a very few) composite numbers. The first composite number for which it is true is 341 = 11 × 31.
Checking a number p by this method is called a primality test because it allows you to quickly find out if a number is composite. The same method works with other bases in place of 2, as long as the base and p have no common factors. With base 2 you just have to check if p is odd or even. 341 "fools the test" for base 2 because it gives the same result a prime number would: 2340 is 1 modulo 341. However, it doesn't fool the test for base 3: 3340 is 56 modulo 341. See 561 for more.
341 is also 143 backwards.
360 can be evenly factored in 24 different ways — more than any other number of this size (see this entry for more about this).
360 is the number of degrees in a full circle. This size for angular division was chosen by the Babylonians because it is 602/10 and because it is close to the distance the sun moves through the zodiac each day (in other words, it is close to the length of the tropical year. In N days, the sun moves just about N degrees across the zodiac. See also 3600 and 3603600.
Often used in quick and sloppy calculations as the number of days in a year. 365 is 52 weeks plus one day.
The length of the mean tropical year, in mean solar days. The tropical year is the amount of time for the Earth to complete one orbit around the Sun, measured from equinox to equinox or from solstice to solstice. It differs from the sidereal year because of precession and the effects of other planets. This is the most important definition of "year" for most people because it is the one that relates the frequency of day and night to the frequency of the seasons. It is the value that solar and lunisolar calendars try to approximate. The figure given here is for the year 2000; it decreases by 0.00000006162 each year due to the slowing of Earth's rotation.
Conveniently for us Earthlings, it happens to be pretty darn close to 360, a number with lots of factors. This means that, for lots of fairly common divisions (like 1/3 of a year, 1/4 of a year, etc.) it's pretty easy to remember how many days is involved because it's so close to what you'd get if the number were exactly 360. Thus: a quarter-year is just about 90 days; etc.
This is the approximation that the Gregorian calendar uses for the number of days in a year.
The length of the tropical year under the Julian calendar. This approximation was probably first known to the Egyptians, who had stuck to a calendar of 365-day years (with no leap-years) for a staggeringly long period of time, and were therefore able to measure how inaccurate it was through observation over time (see 1508.0833, the Sothic cycle).
The length of a sidereal year in mean solar days. A sidereal year is the amount of time it takes Earth to complete one orbit when viewed from a nonrotating reference frame (which is usually approximated by averaging together the positions of many distant stars). It differs from the tropical year because of precession.
See also 365.2568983.
An approximate value of the sidereal year at the time of Carl Friedrich Gauss' work on orbital mechanics and regression analysis. It persists as part of the definition of the astronomical unit.
A reader wrote to me and asked (roughly) the following:
What are the odds of seeing the digits 3, 6 and 8 (in any order, but contiguous) on an automobile license plate? If I see 1000 plates during my drive to work, it is unusual to see 10 cars that feature one of the combinations 368, 386, 638, 683, 836 or 863?
Of course, the answer depends a lot on how many license plates have digits at all, and whether there are lots of plates with 4 or more digits in them, etc. You have to consider how many cars there are with each different number of digits, and figure the odds for each.
For plates with 3-digit numbers, the answer is easy: 6 out of 1000. In general, the odds of seeing your N special digits in an N-digit number are N!/10N. (This is assuming the special digits are different from each other. If some of them are the same, the odds go down.)
For plates with more than N digits, the odds are a little more complicated to compute. For example, for 3 digits in a 4-digit number, the odds are 114 out of 10000. This is 6 less than 120/10000, which is what you get if you take the 6/1000 odds for the 3-digit case and double it. The reason is that you get:
- 60 cases with the 3 digits at the beginning of the 4-digit number
- 60 cases with the 3 digits at the end
- minus 6 cases that got counted twice because the first three are a match and the last three are also a match. These are the 6 cases that consist entirely of the 3 special digits, and have the first and last digit the same: 3683, 3863, 6386, 6836, 8368 and 8638.
The general formula for the N+1 case is (2×10-1)N!/10N
As you would expect, the formula keeps getting more complicated. For the N+2 case it's ((3×10-2)×10-1)N!/10N.
Back to page 9 . . . Forward to page 11