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23

The first prime that is a sum of three consecutive primes (5+7+11) (thanks to Philip Hassey for this tip)

The smallest number that requires more than two terms to express in the form X = 2^a3^b + 2^c3^d + 2^e3^f + ... See also 431

A cult number, noted particularly for associations with the Illuminati and conspiracy theories.

23.140692632779269005...

This is e^π, also called Gelfond's constant. It is notable because if you raise it to the power of a square root, you often get an answer which is very near an integer. See 262537412640768743.999999... for more.

e^π also happens to be kind of close to 20 + π. The actual difference is 19.999099979189475767... Those two sets of 9's are the reason why there are two sets of digits that coincide:

23.140692632779269005... = e^π
  3.141592653589793238... = π

e^{π √5} = 2⁶(Φ)⁶ - 24  +  0.24... (W2)
e^{π √6} = 2⁶(1+√2)⁴ + 24  +  0.12... (W2)
e^{π √6} = (4√3)⁴ - 106  +  0.0091... (R2)
e^{π √10} = 2⁶(Φ)¹² + 24  +  0.013... (W2)
e^{π √10} = 12⁴ - 104  +  0.21... (R2)
e^{π √11} = 32³ + 738  +  0.14... (H1)
e^{π √13} = 2⁶((3+√13)/2)⁶ - 24  +  0.0033... (W2)
e^{π √13} = (12√2)⁴ + 104  +  0.052... (R2)
e^{π √14} = 4⁴(11+8√2)² - 104  +  0.034... (R4)
e^{π √15} = 3³(Φ)²(5+4√5)³ + 745  -  0.022... (H2)
e^{π √16} = 66³ - 744  -  0.68... (H1)
e^{π √19} = 96³ + 744  -  0.22... (H1)
e^{π √20} = 2³(25+13√5)³ - 744  +  0.15... (H2)
e^{π √22} = 2⁶(1+√12)¹² + 24  -  0.00011... (W2)
e^{π √22} = (12√11)⁴ - 104  +  0.0017... (R2)
e^{π √24} = 12³(1+√2)²(5+2√2)³ - 744  +  0.040... (H2)
e^{π √28} = 255³ - 744  -  0.011... (H1)
e^{π √30} = (4√3)⁴(5+4√2)⁴ - 104  +  0.00014... (R4)
e^{π √34} = 12⁴(4+√17)⁴ - 104  +  0.000048... (R4)
e^{π √35} = 16³(15+7√5)³ + 744  +  0.0016... (H2)
e^{π √37} = 2⁶(6+√37)⁶ - 24  +  0.0000013... (W2)
e^{π √37} = (84√2)⁴ + 104  +  0.000021... (R2)
e^{π √40} = 6³(65+27√5)³ - 744  +  0.00046... (H2)
e^{π √42} = 4⁴(21+8√6)⁴ - 104  -  0.0000062... (R4)
e^{π √43} = 960³ + 744  -  0.00022... (H1)
e^{π √46} = 12⁴(147+104√2)² - 104  +  0.0000024... (R4)
e^{π √51} = 48³(4+√17)²(5+√17)³ + 744  -  0.000035... (H2)
e^{π √52} = 30³(31+9√13)³ - 744  +  0.000028... (H2)
e^{π √58} = 2⁶((5+√29)/2)¹² + 24  -  0.000000011... (W2)
e^{π √58} = 396⁴ - 104  +  0.00000017... (R2)
e^{π √67} = 5280³ + 744  -  0.0000014... (H1)
e^{π √70} = (12√7)⁴(5√5+8√2)⁴ - 104  +  0.000000016... (R4)
e^{π √78} = (4√3)⁴(75+52√2)⁴ - 104  +  0.0000000038... (R4)
e^{π √82} = 12⁴(51+8√41)⁴ - 104  +  0.0000000019... (R4)
e^{π √88} = 60³(155+108√2)³ - 744  +  0.000000031... (H2)
e^{π √91} = 48³(227+63√13)³ + 744  +  0.000000019... (H2)
e^{π √102} = (4√3)⁴(200+49√17)⁴ - 104  +  0.000000000072... (R4)
e^{π √115} = 48³(785+351√5)³ + 744  +  0.00000000046... (H2)
e^{π √123} = 480³(32+5√41)²(8+√41)³ + 744  -  0.00000000014... (H2)
e^{π √130} = 12⁴(323+40√65)⁴ - 104  -  0.0000000000012... (R4)
e^{π √142} = 12⁴(467539+330600√2)² - 104  +  0.00000000000024... (R4)
e^{π √148} = 60³(2837+468√37)³ - 744  +  0.0000000000049... (H2)
e^{π √163} = 640320³ + 744  -  0.00000000000075... (H1)
e^{π √187} = 240³(3451+837√17)³ + 744  +  0.000000000000043... (H2)
e^{π √190} = (12√19)⁴(481+340√2)⁴ - 104  +  0.00000000000000068... (R4)
e^{π √232} = 30³(140989+26163√29)³ - 744  +  0.00000000000000032... (H2)
e^{π √235} = 528³(8875+3969√5)³ + 744  +  0.00000000000000023... (H2)
e^{π √267} = 240³(500+53√89)²(625+53√89)³  +  744 - 0.000000000000000010... (H2)
e^{π √403} = 240³(2809615+779247√13)³ + 744  +  0.000000000000000000000080... (H2)
e^{π √427} = 5280³(236674+30303√61)³ + 744  +  0.000000000000000000000012... (H2)

H: Hilbert class polynomial; W: Weber class polynomial; R: Ramanujan class polynomial; number gives class.

24

A highly composite number.

Factorials

24 is a factorial: the product of consecutive integers starting with 1: 24 = 1×2×3×4. This is written "4!". The factorials are: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, ... (Sloane's sequence A000142).

The factorial can be expanded to real numbers in general (and even imaginary and complex numbers) by using the Gamma function.

25

Squares

25 is a "perfect square", the product of an integer with itself, 5 times 5. It is called a "square" because 25 things can be arranged in a square pattern using 5 rows with 5 objects in each row.

The Pythagorean Theorem states that if a triangle has a right angle, and the side opposite the angle is length C, then A² + B² = C². 25 is a square, 5², and is the sum of two squares: 3² + 4² = 5². This is the simplest example of a square of an integer which is the sum of two other integer squares and corresponds to the 3-4-5 right triangle, which is used as an example in most ancient texts describing the Pythagorean theorem. There are many "Pythagorean triples" such as 3, 4, and 5.

25 is also a square that is the sum of two consecutive squares. There aren't many of these; see 841 for more. See also 216 and 143.

25 and 27 are the only case of a square and a cube separated by 2. I have seen a proof of this but it is difficult to understand, so I cannot confirm it.

25₁₀=31{8}. Some calculators have labeled "OCT" for "octal" and "DEC" for "decimal", inviting the following nerdy joke:

Why do programmers mistake Halloween for Christmas?
*Because 31 OCT equals 25 DEC."

See also 69.

26

This number has special divine significance in gematria because it is the sum of the letter-values of the Hebrew biblical name of God: Yod + Heh + Vau + Heh = 10 + 5 + 6 + 5 = 26.

27

The third cube, 3³.

27!+1 is prime. There aren't many numbers N such that N!+1 is prime; and it's hard to find them because factorials are so large. As of May 2002, the only known values of N are: 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, ... (Sloane's A002981).

27×2²⁷+2²⁷+27 is prime.

27 items can be arranged into a hexagon-like arrangement with alternating sides of 3 and 4 items. So, I call it the "3¹/₂^th hexagonal number":
o o o o o o o o o o o o o o o o o o o o o o o o o o o
Because 27 is the smallest factor of 999 that is not also a factor of 9 or 99, 27 is the smallest number whose reciprocal has a 3-digit repeating pattern. The next is 37. Because 27×37=999, we also have the nice relationship 1/27 = .037037037..., and 1/37 = .027027027... See also 239 and 757.

The digits of 27, 2 and 7, plus the numbers in between, add up to 27: 2+3+4+5+6+7=27. The only other 2-digit number that shares this property is 15. (And their sum is 42!)

To test a number for divisibility by 27:

• 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.
• Check the resulting 3-digit number for divisibility by 9. If it isn't, the original number isn't divisible by 27.
• If it is, divide it by 9 (see the 9 entry for a simple way to do this). Then check the answer for divisibility by 3. If it is, the original number is divisible by 27, otherwise it isn't.

See also 89, 134217728, 10888869450418352160768000000 and 10^{3.0056206947796095239×1029}.

27 is the largest number for which the digits of its cube is equal to the number: 27³ = 19683, 1 + 9 + 6 + 8 + 3 = 27. The smallest number with this property is 8, and it is perhaps of interest that 8 and 27 are themselves cubes.

In Hindu astrology there are 27 nakshatra, (which means "stars"), each controls a section of the zodiac that is 13^o20' wide; the first few are called ashwini, bharani, krittika. 27 is the closest integer approximation to the sidereal month, although most cultures with lunar Zodiac divisions tend to use 28 instead. It is also a rather nice coindidence that the size of the sun and moon in the sky (about half a degree, or 1/720 of the full circle) is almost exactly 1/27 of the distance covered by the moon each day — in other words, the fraction of the moon's size in proportion to a nakshatra is equal to the fraction of one nakshatra in proportion to the entire zodiac.

There are 27 different ways that three people can throw a choice in three-player rock-paper-scissors²³. In this variant by Paul Hsieh, the rules are a little more elaborate. As in normal rock-paper-scissors²⁴, the hand gestures are the closed fist (rock), all fingers out flat (paper) and two fingers out like a "victory" or "peace" sign (scissors). If all three players throw the same thing (a dead draw) or if they throw all three different signs (a loop or vicious circle) there is no winner and they all play again; there are 9 ways this can happen. If two players throw the same sign, the third player is the odd one out. If the odd one's sign beats the other two players' sign (a slam) then the odd one wins instantly (there are 9 ways this can happen). Otherwise, (the last 9 possibilities) the odd one's sign loses to the other two (a wipeout); the odd one is eliminated and the other two players compete using a normal two-player match.

There are also 27 gambits in Professional Rock-Paper-Scissors²⁵; a gambit is a sequence of three consecutive throws, used as a component unit of a match. (Players go so fast that they need to practice sequences of throws many times and often use memorized gambits as a way to gain the upper hand against less-experienced players who can't keep up with the pace.)

27 is a psychologically random number, similar to 17 and 37 and having no particular cultural origin. Like 37, it is often used when some random-sounding large number is needed. For example, in Graham Greene's 1953 play The Living Room one finds the line:
ROSE Since my last confession three weeks ago I've committed adultery twenty-seven times.
(27 is also my favorite cult number, for various reasons, for example it's the street number of a house where I grew up, and my age was 27 years + 27 days when I met a certain close friend.)

27.212220817

Length of the mean draconic (or nodical) month, in mean solar days. The draconic month is the amount of time for the Moon to complete one orbit around the Earth, measured between rising node crossings; the rising node is when the Moon crosses the plane of Earth's orbit. This period is significant because it determines when eclipses are possible. This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure increases by 0.000000003833 each year.

27.321582241

Length of the mean tropical month, in mean solar days. The tropical month is the amount of time for the Moon to complete one orbit around the Earth in relation to the equinox and solstice points (which shift relative to the stars because of precession). This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure increases by 0.000000001506 each year.

27.321661547

Length of the mean sidereal month, in mean solar days. The sidereal month is the amount of time for the Moon to complete one orbit around the Earth in relation to the stars (and thus, Zodiac signs). This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure increases by 0.000000001857 each year.

27.554549878

Length of the mean anomalistic month, in mean solar days. The anomalistic month is the amount of time for the Moon to complete one orbit around the Earth, measured from perigee to perigee. The alignment of this cycle with other cycles (most notably the synodic month) affects the timing of the moon's phases (see here for more). This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure decreases by 0.000000010390 each year.

28

A perfect number, one that is the sum of its divisors: 28=1+2+4+7+14. The perfect numbers are: 6, 28, 496, 8128, 33550336, ... (Sloane's A000396). See here for a complete list. See here for more about perfect numbers.

28 is used as an approximation to the number of days in the lunar month, in those cultures that need such a thing. For example, in Chinese astrology there are 28 "lunar mansions", divisions of the zodiac analagous to the 12 houses of the sun (called chuen, k'ang, ti, fang, etc.). In old Arabian astrology they are called manazils (the first few are alnath, albotain, azoraya). In some versions of Hindu/Indian astrology these 28 mansions are called nakshatra (Other versions have 27 nakshatra; the versions with 28 interpolate the mansion abhijit between uttarashadha and shravana).

29

29 is the lowest base with 'easy' divisibility tests for 12 different numbers, assuming that the casting out 11's method is allowed. In base 29 you can test for divisibility by 2, 3, 4, 5, 6, 7, 10, 14, 15, 28, 29 and 30.

Here is a list of record-setters for this property (bases with high numbers of testable divisors). The divisors in bold are tested just by looking at the last digit; the divisors in the plain font are tested by the digit-addition technique (casting out 9's), and the divisors in italic are tested by alternate addition and subtraction (see the entry on 11 for a description):

If you want the record-setters for prime divisors only, check the entry for 14. If you don't think the casting out 11's method should count, see the entry for 21.

29.530588235... = 25101/850

The best approximation to the length of the synodic month that can be had with an integer fraction. Better approximations are of little use because the moon is slowing down in its orbit, and before the approximation drifts enough to matter, the moon will have slowed down enough to make the approximation obsolete.

29.530588853

Length of the mean synodic month, in mean solar days. The synodic month is the amount of time for the Moon to complete one orbit around the Earth, measured from new moon to new moon — the common "lunar month" used for calendar purposes. This is the value for the year 2000; because of the effects of other planets the Moon's orbit is changing, and this figure increases by 0.000000002162 each year.

29.5305941358 = 29+¹²/₂₄+⁷⁹³/_(24×1080) = 29+³¹/₆₀+⁵⁰/_60²+⁸/_60³+²⁰/_60⁴ = 765433/25920

The approximation to the length of the synodic month developed by the Chaldeans of Babylonia, and still used today in calculation of the Hebrew calendar. The first sum, 29+¹²/₂₄+⁷⁹³/_(24×1080), shows how the value is expressed in Hebrew Talmudic time units; the other sum is the Sumerian sexagesimal fraction. It is longer than the real synodic month by enough to amount to one day every 15000 years. See also 689472, 1969920.

30

30=2×3×5, a primorial. It is sometimes used as a highly composite number although it is not a record-setter (24 has as many factors and 36 has one more).

30 is the closest integer to the length of the synodic month. Because of this, (and probably also related to its high number of factors, and the Sumerian base 60), there have been many calendars that use 30-day months. 30 is also used as the base of a Hindu time division system that includes several successive powers of 30, see 405000.

30.48

By agreement, the exact number of centimeters in a foot (see 2.54).

See also 1609.344.

31

31 is 2⁵-1, a Mersenne number and a Mersenne prime. It is also a member of the "primeth number" sequence A_n = {1, 2, 3, 5, 11, 31, ...} where A_n is the A_n-1^th prime. See 127 for more on both of these topics.

32

8=2³ and 9=3² are fairly close; 2⁴ and 4² are equal to each other. The next-closest pair of this type is 25=5² and 32=2⁵.

33

A palindrome is a number, word, sentence etc. that is the same when its digits/letters are reversed. 33 is a palindrome in binary as well as in base 10. There are very few numbers with this property; the sequence starts:

3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15351, 32223, 39993, 53235, 53835, 73737, 585585, ... (Sloane's integer sequence A007632) (Charlton Harrison)

34

The sum of the rows, columns, and diagonals of a 4x4 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(N²+1)/2 or (N+N³)/2. That sequence runs: 1, 5, 15, 34, 65, 111, 175, ... (although there are no 2x2 magic squares). This is Sloane's sequence A006003.

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.

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 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).

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