6 (number)

6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number; its proper divisors are 1, 2 and 3.

Since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and $\mathcal{S}$ -perfect number.^[1]^[2]

As a perfect number:

6 is related to the Mersenne prime 3, since 2¹(2² − 1) = 6. (The next perfect number is 28.)
6 is the only even perfect number that is not the sum of successive odd cubes.^[3]
As a perfect number, 6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one number; the square number, 25.

Six is the only number that is both the sum and the product of three consecutive positive numbers.^[4]

Unrelated to 6 being a perfect number, a Golomb ruler of length 6 is a "perfect ruler."^[5] Six is a congruent number.

Six is the first discrete biprime (2 × 3) and the first member of the (2 × q) discrete biprime family.

Six is the smallest natural number that can be written as the sum of two positive rational cubes which are not integers: $6 = \left(\frac{17}{21}\right)^3 + \left(\frac{37}{21}\right)^3.$ Others up to 100 are: 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98. (sequence A228499 in OEIS)

Six is a unitary perfect number, a harmonic divisor number and a superior highly composite number, the last to also be a primorial. The next superior highly composite number is 12. The next primorial is 30.

5 and 6 form a Ruth-Aaron pair under either definition.

There are no Graeco-Latin squares with order 6. If n is a natural number that is not 2 or 6, than there is a Graeco-Latin square with order n.

The smallest non-abelian group is the symmetric group S₃ which has 3! = 6 elements.

S₆, with 720 elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4 and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number n for which there is a construction of n isomorphic objects on an n-set A, invariant under all permutations of A, but not naturally in one-to-one correspondence with the elements of A. This can also be expressed category theoretically: consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n = 6.

6 similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.

6 is the largest of the four all-Harshad numbers.

A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of a power of 2 (namely 2¹) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon.

Six is also an octahedral number. It is a triangular number and so is its square (36).

There are six basic trigonometric functions.

There are six convex regular polytopes in four dimensions.

Six is the four-bit binary complement of number nine (6 = 0110₂ and 9 = 1001₂).

The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.

All primes above 3 are of the form 6n ± 1 for n ≥ 1.

In numeral systems

Base	Numeral system	Representation
2	binary	110
3	ternary	20
4	quaternary	12
5	quinary	11
6	senary	10
over 6 (decimal, duodecimal, hexadecimal, etc.)		6

In bases 10, 15 and 30, 6 is a 1-automorphic number.

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15	16	17	18	19	20		21	22	23	24	25		50	100	1000
6 × x	6	12	18	24	30	36	42	48	54	60		66	72	78	84	90	96	102	108	114	120		126	132	138	144	150		300	600	6000

Division	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
6 ÷ x	6	3	2	1.5	1.2	1	0.857142	0.75	0.6	0.6		0.54	0.5	0.461538	0.428571	0.4
x ÷ 6	0.16	0.3	0.5	0.6	0.83	1	1.16	1.3	1.5	1.6		1.83	2	2.16	2.3	2.5

Exponentiation	1	2	3	4	5	6	7	8	9	10		11	12	13
6^x	6	36	216	1296	7776	46656	279936	1679616	10077696	60466176		362797056	2176782336	13060694016
x⁶	1	64	729	4096	15625	46656	117649	262144	531441	1000000		1771561	2985984	4826809

Greek and Latin word parts

X-ray of a polydactyl human hand with six fingers

Hexa

Hexa is classical Greek for "six". Thus:

"Hexadecimal" combines hexa- with the Latinate decimal to name a number base of 16
A hexagon is a regular polygon with six sides
- L’Hexagone is a French nickname for the continental part of Metropolitan France
A hexahedron is a polyhedron with six faces, with a cube being a special case
Hexameter is a poetic form consisting of six feet per line
A "hex nut" is a nut with six sides, and a hex bolt has a six-sided head
The prefix "hexa-" also occurs in the systematic name of many chemical compounds, such as "hexamethyl"

The prefix sex-

Sex- is a Latin prefix meaning "six". Thus:

Senary is the ordinal adjective meaning "sixth"
People with sexdactyly have six fingers on each hand
The measuring instrument called a sextant got its name because its shape forms one-sixth of a whole circle
A group of six musicians is called a sextet
Six babies delivered in one birth are sextuplets
Sexy prime pairs - Prime pairs differing by six are sexy, because sex is the Latin word for six.^[6]

Evolution of the glyph

The evolution of our modern glyph for 6 appears rather simple when compared with that for the other numerals. Our modern 6 can be traced back to the Brahmins of India, who wrote it in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Ghubar Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.^[7]

On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a 'b' is not practical.

Just as in most modern typefaces, in typefaces with text figures the 6 character usually has an ascender, as, for example, in .

This numeral resembles an inverted 9. To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels.

In music

A standard guitar has 6 strings

In artists

Les Six ("The Six" in English) was a group consisting of the French composers Georges Auric, Louis Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc and Germaine Tailleferre in the 1920s
Bands with the number six in their name include Six Organs of Admittance, 6 O'Clock Saints, Electric Six, Eve 6, Los Xey (sei is Basque for "six"), Out On Blue Six, Six In Six, Sixpence None the Richer, Slant 6, Vanity 6, and You Me At Six
#6 is the pseudonym of American musician Shawn Crahan, when performing with the band Slipknot

Also the number of Tuka's album "Life Death Time Eternal" debuting on the Aria Charts.

In instruments

A standard guitar has 6 strings
Most woodwind instruments have 6 basic holes or keys (e.g., bassoon, clarinet, pennywhistle, saxophone); these holes or keys are usually not given numbers or letters in the fingering charts

In music theory

There are 6 whole tones in an octave.
There are 6 semitones in a tritone.

In works

"Six geese a-laying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas."
Divided in six arias, Hexachordum Apollinis is generally regarded as one of the pinnacles of Johann Pachelbel's oeuvre.
The concerti grossi Opus 3, organ concertos Opus 4 and Opus 7 (each) by Georg Frideric Handel.
The theme of the sixth album by Dream Theater, Six Degrees Of Inner Turbulence, was the number six: the album has six songs, and the sixth song — that is, the complete second disc — explores the stories of six individuals suffering from various mental illnesses.

In religion

There are 6:
- Points on a Star of David
- Orders of the Mishnah
- Symbolic foods placed on the Passover Seder Plate
Yahweh took 6 days to create the world in the Old Testament Book of Genesis; humankind was created on day 6. In the City of God, Augustine of Hippo suggested (book 11, chapter 30) that God's creation of the world took six days because 6 is a perfect number.
The Jewish holiday of Shavuot starts on the sixth day of the Hebrew month of Sivan
Seraphs have six wings.
In Islam:
- There are Six articles of belief
- Fasting six days of Shawwal, together with the month of Ramadan, is equivalent to fasting the whole year
In Hindu theology, a trasarenu is the combination of 6 celestial paramānus (atoms)

In science

Astronomy

Messier object M6, a magnitude 4.5 open cluster in the constellation Scorpius, also known as the Butterfly Cluster
The New General Catalogue object NGC 6, a spiral galaxy in the constellation Andromeda
The Saros number of the solar eclipse series which began on March 16, 2691 BC and ended on May 3, 1393 BC. The duration of Saros series 6 was 1298.1 years, and it contained 73 solar eclipses
The Saros number of the lunar eclipse series which began on July 25, 2642 BC and ended on February 10, 1091 BC. The duration of Saros series 6 was 1550.6 years, and it contained 87 lunar eclipses
The Roman numeral VI:
- Stands for subdwarfs in the Yerkes spectral classification scheme
- (Usually) stands for the sixth-discovered satellite of a planet or minor planet (e.g. Jupiter VI)

Biology

The cells of a beehive are 6-sided
Insects have 6 legs

Chemistry

A benzene molecule has a ring of 6 carbon atoms
6 is the atomic number of carbon
The sixfold symmetry of snowflakes arises from the hexagonal crystal structure of ordinary ice
A hexamer is an oligomer made of six subunits

The cells of a beehive are 6-sided

Medicine

There are 6 tastes in traditional Indian Medicine called Ayurveda: sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body
Phase 6 is one of six pandemic influenza phases

Physics

In the Standard Model of particle physics, there are 6 types of quark and 6 types of lepton

In the Standard Model of particle physics, there are 6 types of quarks and 6 types of leptons
In statistical mechanics, the six-vertex model has 6 possible configurations of arrows at each vertex

In sports

6 entities in a grouping:
- The National Basketball Association has six divisions (the National Hockey League had six divisions until realigning into four divisions for the 2013-14 season)
- The Original Six teams in the National Hockey League are Toronto, Chicago, Montreal, New York, Boston, and Detroit. They are the oldest remaining teams in the league, though not necessarily the first six; they comprised the entire league from 1942 to 1967
Number of players:
- Six-man football is a variant of American or Canadian football, played by smaller schools with insufficient enrollment to field the traditional 11-man (American) or 12-man (Canadian) squad
- In football (soccer), the number of substitutes combined by both teams, that are allowed in the game.
- In ice hockey, the number of players per team, including the goaltender, that are on the ice at any one time, excluding penalty situations.
- In volleyball, 6 players from each team on each side play against each other
Scoring and related terminology:
- In some sports, 6 goals is known as a double-hat-trick, but is very hard to accomplish
- In American and Canadian football, a touchdown earns 6 points
- In Australian Rules football, six points are received for a goal
- In baseball scorekeeping, the number 6 represents the shortstop's position.
- In cricket (See cricket terminology):
  - a "six" or "sixer" is a shot in which the ball clears the boundary without bouncing, scoring six runs
  - there are six balls to an over
Jersey number 6 and car number 6:
- In football/soccer:
  - Is the number of AC Milan's centre back and captain Franco Baresi; the shirt was retired in 1997
- In NASCAR:
  - The number 6 is currently owned by Roush Fenway Racing. Since the 2007 season, the first year in which Roush Racing was merged with the Fenway Sports Group that owns the Boston Red Sox, the Cup Series version of the car has been driven by David Ragan. From 1988 to 2006, Mark Martin drove the #6 in the Cup Series for what was then Roush Racing.
- In rugby:
  - In most rugby league competitions (but not the European Super League, which uses static squad numbering), the jersey number 6 is worn by the starting five-eighth (Southern Hemisphere term) or stand-off half (Northern Hemisphere term).
  - In rugby union, the starting blindside flanker wears jersey number 6. (Some teams use "left" and "right" flankers instead of "openside" and "blindside", with 6 being worn by the starting left flanker.)
- The jersey number 6 has been retired by several North American sports teams in honor of past playing greats or other key figures (or, in two cases, a team's fans):
  - In Major League Baseball:
    - The Atlanta Braves, for manager Bobby Cox
    - The Boston Red Sox, for Johnny Pesky
    - The Detroit Tigers, for Hall of Famer Al Kaline
    - The Minnesota Twins, for Tony Oliva
    - The New York Yankees, for manager Joe Torre
    - The St. Louis Cardinals, for Hall of Famer Stan Musial
    - The San Diego Padres, for Steve Garvey
  - In the NBA:
    - The Boston Celtics, for Hall of Famer Bill Russell
    - The Orlando Magic, for their fans (the "sixth man")
    - The Philadelphia 76ers, for Hall of Famer Julius Erving
    - The Phoenix Suns, for Walter Davis
    - The Sacramento Kings, also for their fans
  - In the NFL:
    - The Kansas City Chiefs, for Warren McVea
  - In the NHL:
    - The Detroit Red Wings, for Larry Aurie
    - The Pittsburgh Penguins, for Ian Ackerman
    - The Toronto Maple Leafs, for Hall of Famer Ace Bailey. The Leafs have a unique policy of not retiring numbers unless the player honoured either died or suffered a career-ending incident while a member of the team. Bailey suffered a fractured skull during a game in 1933; while he recovered and lived for nearly 60 years after the incident, he never played again. The Leafs would issue the number to Ron Ellis in 1968 at Bailey's personal request, and Ellis wore it until his own retirement in 1981.

In technology

On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry it is the key for J and K, and on the BlackBerry 8700 series and Curve 8900 with full keyboard, it is the key for F
The "6-meter band" in amateur radio includes the frequencies from 50 to 54 MHz
6 is the resin identification code used in recycling to identify polystyrene

In calendars

In the ancient Roman calendar, Sextilis was the sixth month. After the Julian reform, June became the sixth month and Sextilis was renamed August
Sextidi was the sixth day of the decade in the French Revolutionary calendar

In the arts and entertainment

The Super 6 - 1966 animated cartoon series featuring six different super-powered heroes.
The Bionic Six are the heroes of the eponymous animated series
The number of sides on a cube, hence the highest number on a standard die
The highest number on one end of a standard domino
The Power of Six is a book written by Pittacus Lore, and the second in the Lorien Legacies series.
Number Six (Tricia Helfer), is a family of fictional characters from the reimagined science fiction television series, Battlestar Galactica
Number 6 is a character in the book series Lorien Legacies
Number 6 (Teresa Palmer) is a character in the movie I Am Number Four (2011)
Number 6, the main protagonist in The Prisoner played by Patrick McGoohan, portrayed by Jim Caviezel in the remake.
Six is a character in the television series Blossom played by Jenna von Oÿ.

In other fields

International maritime signal flag for 6

The standard term in office for a United States senator is six years
The name of the smallest group of Cub Scouts, traditionally consisting of six people and is led by a 'sixer'.
Six pack is a common form of packaging for six bottles or cans of drink (especially beer), and by extension, other assemblages of six items.
The number of fundamental flight instruments lumped together on a cockpit display, called the six-pack.
The number of feet below ground level a coffin is traditionally buried; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead
In Astrology, Virgo is the 6th astrological sign of the Zodiac
The number of dots in a Braille cell.
The number of Royal Stickers that Mario must collect in the Nintendo 3DS game Paper Mario: Sticker Star
There are said to be no more than six degrees of separation between any two people on Earth. See also Six degrees (disambiguation).
Extra-sensory perception is sometimes called the "sixth sense"
Six Cardinal Directions: north, south, east, west, up, and down
The Six Dynasties form part of Chinese history
6 is a lucky number in Chinese culture
The Birmingham Six were held in prison for 16 years
"Six" is used as an informal slang term for the British Secret Intelligence Service, MI6
Six Flags amusement parks and theme parks

References

↑ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 11. ISBN 978-1-84800-000-1.
↑ "Granville number". OeisWiki. The Online Encyclopedia of Integer Sequences. Archived from the original on 29 March 2011. Retrieved 27 March 2011.
↑ David Wells, The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books (1987): 67
↑ Peter Higgins, Number Story. London: Copernicus Books (2008): 12
↑ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 72
↑ Chris K. Caldwell; G. L. Honaker Jr. (2009). Prime Curios!: The Dictionary of Prime Number Trivia. CreateSpace Independent Publishing Platform. p. 11. ISBN 978-1448651702.
↑ Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.66

The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66—68
A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links ISBN 0-521-42385-6
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 67 - 69

External links

Integers

0s

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

100s

100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119

120 121 122 123 124 125 126 127 128 129

130 131 132 133 134 135 136 137 138 139

140 141 142 143 144 145 146 147 148 149

150 151 152 153 154 155 156 157 158 159

160 161 162 163 164 165 166 167 168 169

170 171 172 173 174 175 176 177 178 179

180 181 182 183 184 185 186 187 188 189

190 191 192 193 194 195 196 197 198 199

200s

200 201 202 203 204 205 206 207 208 209

210 211 212 213 214 215 216 217 218 219

220 221 222 223 224 225 226 227 228 229

230 231 232 233 234 235 236 237 238 239

240 241 242 243 244 245 246 247 248 249

250 251 252 253 254 255 256 257 258 259

260 261 262 263 264 265 266 267 268 269

270 271 272 273 274 275 276 277 278 279

280 281 282 283 284 285 286 287 288 289

290 291 292 293 294 295 296 297 298 299

300s

300 301 302 303 304 305 306 307 308 309

310 311 312 313 314 315 316 317 318 319

320 321 322 323 324 325 326 327 328 329

330 331 332 333 334 335 336 337 338 339

340 341 342 343 344 345 346 347 348 349

350 351 352 353 354 355 356 357 358 359

360 361 362 363 364 365 366 367 368 369

370 371 372 373 374 375 376 377 378 379

380 381 382 383 384 385 386 387 388 389

390 391 392 393 394 395 396 397 398 399

400s

400 401 402 403 404 405 406 407 408 409

410 411 412 413 414 415 416 417 418 419

420 421 422 423 424 425 426 427 428 429

430 431 432 433 434 435 436 437 438 439

440 441 442 443 444 445 446 447 448 449

450 451 452 453 454 455 456 457 458 459

460 461 462 463 464 465 466 467 468 469

470 471 472 473 474 475 476 477 478 479

480 481 482 483 484 485 486 487 488 489

490 491 492 493 494 495 496 497 498 499

500s

500 501 502 503 504 505 506 507 508 509

510 511 512 513 514 515 516 517 518 519

520 521 522 523 524 525 526 527 528 529

530 531 532 533 534 535 536 537 538 539

540 541 542 543 544 545 546 547 548 549

550 551 552 553 554 555 556 557 558 559

560 561 562 563 564 565 566 567 568 569

570 571 572 573 574 575 576 577 578 579

580 581 582 583 584 585 586 587 588 589

590 591 592 593 594 595 596 597 598 599

600s

600 601 602 603 604 605 606 607 608 609

610 611 612 613 614 615 616 617 618 619

620 621 622 623 624 625 626 627 628 629

630 631 632 633 634 635 636 637 638 639

640 641 642 643 644 645 646 647 648 649

650 651 652 653 654 655 656 657 658 659

660 661 662 663 664 665 666 667 668 669

670 671 672 673 674 675 676 677 678 679

680 681 682 683 684 685 686 687 688 689

690 691 692 693 694 695 696 697 698 699

700s

700 701 702 703 704 705 706 707 708 709

710 711 712 713 714 715 716 717 718 719

720 721 722 723 724 725 726 727 728 729

730 731 732 733 734 735 736 737 738 739

740 741 742 743 744 745 746 747 748 749

750 751 752 753 754 755 756 757 758 759

760 761 762 763 764 765 766 767 768 769

770 771 772 773 774 775 776 777 778 779

780 781 782 783 784 785 786 787 788 789

790 791 792 793 794 795 796 797 798 799

800s

800 801 802 803 804 805 806 807 808 809

810 811 812 813 814 815 816 817 818 819

820 821 822 823 824 825 826 827 828 829

830 831 832 833 834 835 836 837 838 839

840 841 842 843 844 845 846 847 848 849

850 851 852 853 854 855 856 857 858 859

860 861 862 863 864 865 866 867 868 869

870 871 872 873 874 875 876 877 878 879

880 881 882 883 884 885 886 887 888 889

890 891 892 893 894 895 896 897 898 899

900s

900 901 902 903 904 905 906 907 908 909

910 911 912 913 914 915 916 917 918 919

920 921 922 923 924 925 926 927 928 929

930 931 932 933 934 935 936 937 938 939

940 941 942 943 944 945 946 947 948 949

950 951 952 953 954 955 956 957 958 959

960 961 962 963 964 965 966 967 968 969

970 971 972 973 974 975 976 977 978 979

980 981 982 983 984 985 986 987 988 989

990 991 992 993 994 995 996 997 998 999

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