Editing 7 (section)

==In mathematics==
Seven, the fourth prime number, is not only a [[Mersenne prime]] (since <math>2^3 - 1 = 7</math>) but also a [[double Mersenne prime]] since the exponent, 3, is itself a Mersenne prime.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Double Mersenne Number|url=https://mathworld.wolfram.com/DoubleMersenneNumber.html|access-date=2020-08-06|website=mathworld.wolfram.com}}</ref> It is also a [[Newman–Shanks–Williams prime]],<ref>{{Cite web |url=https://oeis.org/A088165 |title=Sloane's A088165 : NSW primes |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}}</ref> a [[Woodall prime]],<ref>{{Cite web |url=https://oeis.org/A050918 |title=Sloane's A050918 : Woodall primes |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}}</ref> a [[factorial prime]],<ref>{{Cite web |url=https://oeis.org/A088054 |title=Sloane's A088054 : Factorial primes |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}}</ref> a [[Harshad number]], a [[lucky prime]],<ref>{{Cite web |url=https://oeis.org/A031157 |title=Sloane's A031157 : Numbers that are both lucky and prime |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}}</ref> a [[happy number]] (happy prime),<ref>{{Cite web |url=https://oeis.org/A035497 |title=Sloane's A035497 : Happy primes |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}}</ref> a [[safe prime]] (the only {{vanchor|Mersenne safe prime}}), a [[Leyland number#Leyland number of the second kind|Leyland number of the second kind]]<ref>{{Cite OEIS|A045575|Leyland numbers of the second kind}}</ref> and [[Leyland number#Leyland number of the second kind|Leyland prime of the second kind]]<ref>{{Cite OEIS|A123206|Leyland prime numbers of the second kind}}</ref> {{nowrap|(<math>2^5-5^2</math>),}} and the fourth [[Heegner number]].<ref>{{Cite web |url=https://oeis.org/A003173 |title=Sloane's A003173 : Heegner numbers |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=2016-06-01}}</ref> Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers.

A seven-sided shape is a [[heptagon]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Heptagon |url=https://mathworld.wolfram.com/Heptagon.html |access-date=2020-08-25 |website=mathworld.wolfram.com}}</ref> The [[Regular polygon|regular]] ''n''-gons for ''n'' ⩽ 6 can be constructed by [[compass and straightedge]] alone, which makes the heptagon the first regular polygon that cannot be directly constructed with these simple tools.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=7 |url=https://mathworld.wolfram.com/7.html |access-date=2020-08-07 |website=mathworld.wolfram.com}}</ref>

7 is the only number ''D'' for which the equation {{nowrap|1=2{{sup|''n''}} − ''D'' = ''x''{{sup|2}}}} has more than two solutions for ''n'' and ''x'' [[Natural number|natural]]. In particular, the equation {{nowrap|1=2{{sup|''n''}} − 7 = ''x''{{sup|2}}}} is known as the [[Ramanujan–Nagell equation]]. 7 is one of seven numbers in the positive [[Quadratic form|definite quadratic]] [[integer matrix]] representative of all [[Parity (mathematics)|odd]] numbers: {1, 3, 5, 7, 11, 15, 33}.<ref>{{Cite book |last1=Cohen |first1=Henri |url=https://link.springer.com/book/10.1007/978-0-387-49923-9 |title=Number Theory Volume I: Tools and Diophantine Equations |publisher=[[Springer Science+Business Media|Springer]] |year=2007 |isbn=978-0-387-49922-2 |edition=1st |series=[[Graduate Texts in Mathematics]] |volume=239 |pages=312–314 |chapter=Consequences of the Hasse–Minkowski Theorem |doi=10.1007/978-0-387-49923-9 |oclc=493636622 |zbl=1119.11001}}</ref><ref>{{Cite OEIS|A116582|Numbers from Bhargava's 33 theorem.|access-date=2024-02-03}}</ref>

There are 7 [[frieze group]]s in two dimensions, consisting of [[symmetry group|symmetries]] of the [[Plane (geometry)|plane]] whose group of [[Translation (geometry)|translations]] is [[isomorphic]] to the group of [[integer]]s.<ref>{{Cite book |last1=Heyden |first1=Anders |url=https://books.google.com/books?id=4yCqCAAAQBAJ&q=seven+frieze+groups&pg=PA661 |title=Computer Vision – ECCV 2002: 7th European Conference on Computer Vision, Copenhagen, Denmark, May 28–31, 2002. Proceedings. Part II |last2=Sparr |first2=Gunnar |last3=Nielsen |first3=Mads |last4=Johansen |first4=Peter |date=2003-08-02 |publisher=Springer |isbn=978-3-540-47967-3 |pages=661 |quote=A frieze pattern can be classified into one of the 7 frieze groups...}}</ref> These are related to the [[17 (number)|17]] [[wallpaper group]]s whose transformations and [[Isometry|isometries]] repeat two-dimensional patterns in the plane.<ref>{{Cite book |author1=Grünbaum, Branko |author-link=Branko Grünbaum |author2=Shephard, G. C. |author2-link=G.C. Shephard |url-access=registration |url=https://archive.org/details/isbn_0716711931 |title=Tilings and Patterns |chapter=Section 1.4 Symmetry Groups of Tilings |publisher=W. H. Freeman and Company |location=New York |year=1987 |pages=40–45 |doi=10.2307/2323457 |jstor=2323457 |isbn=0-7167-1193-1 |oclc=13092426 |s2cid=119730123 }}</ref><ref>{{Cite OEIS |A004029 |Number of n-dimensional space groups. |access-date=2023-01-30 }}</ref>

A heptagon in [[Euclidean space]] is unable to generate [[uniform tiling]]s alongside other polygons, like the regular [[pentagon]]. However, it is one of fourteen polygons that can fill a [[Euclidean tilings by convex regular polygons#Plane-vertex tilings|plane-vertex tiling]], in its case only alongside a regular [[Equilateral triangle|triangle]] and a 42-sided polygon ([[:File:3.7.42 vertex.png|3.7.42]]).<ref>{{Cite journal |first1=Branko |last1=Grünbaum |author-link=Branko Grünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=[[Mathematics Magazine]] |volume=50 |issue=5 |publisher=Taylor & Francis, Ltd. |page=231 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 |archive-date=2016-03-03 |access-date=2023-01-09 |archive-url=https://web.archive.org/web/20160303235526/http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |url-status=dead }}</ref><ref>{{Cite web |last=Jardine |first=Kevin |url=http://gruze.org/tilings/3_7_42_shield|title=Shield - a 3.7.42 tiling |website=Imperfect Congruence |access-date=2023-01-09 }} 3.7.42 as a unit facet in an irregular tiling.</ref> Otherwise, for any regular ''n''-sided polygon, the maximum number of intersecting diagonals (other than through its center) is at most 7.<ref>{{Cite journal |last1=Poonen |first1=Bjorn |author1-link=Bjorn Poonen |last2=Rubinstein |first2=Michael |title=The Number of Intersection Points Made by the Diagonals of a Regular Polygon |url=https://math.mit.edu/~poonen/papers/ngon.pdf |journal=SIAM Journal on Discrete Mathematics |volume=11 |issue=1 |publisher=[[Society for Industrial and Applied Mathematics]] |location=Philadelphia |year=1998 |pages=135–156 |doi=10.1137/S0895480195281246 |arxiv=math/9508209 |mr=1612877  |zbl=0913.51005 |s2cid=8673508 }}</ref>

In two dimensions, there are precisely seven [[Euclidean tilings by convex regular polygons#k-uniform tilings|7-uniform]] ''Krotenheerdt'' tilings, with no other such ''k''-uniform tilings for ''k'' > 7, and it is also the only ''k'' for which the count of ''Krotenheerdt'' tilings agrees with ''k''.<ref>{{Cite OEIS |A068600 |Number of n-uniform tilings having n different arrangements of polygons about their vertices. |access-date=2023-01-09 }}</ref><ref>{{Cite journal |first1=Branko |last1=Grünbaum |author-link=Branko Grünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |journal=[[Mathematics Magazine]] |volume=50 |issue=5 |publisher=Taylor & Francis, Ltd. |page=236 |doi=10.2307/2689529 |jstor=2689529 |s2cid=123776612 |zbl=0385.51006 |archive-date=2016-03-03 |access-date=2023-01-09 |archive-url=https://web.archive.org/web/20160303235526/http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |url-status=dead }}</ref>

The [[Fano plane]], the smallest possible [[finite projective plane]], has 7 points and 7 lines arranged such that every line contains 3 points and 3 lines cross every point.<ref>{{Cite book |first1=Tomaž |last1=Pisanski |first2=Brigitte |last2=Servatius |author1-link=Tomaž Pisanski |author2-link=Brigitte Servatius |title=Configurations from a Graphical Viewpoint |chapter=Section 1.1: Hexagrammum Mysticum |chapter-url=https://link.springer.com/chapter/10.1007/978-0-8176-8364-1_5 |edition=1 |publisher=[[Birkhäuser]] |series=Birkhäuser Advanced Texts |location=Boston, MA |year=2013 |pages=5–6 |isbn=978-0-8176-8363-4 |oclc=811773514 |doi=10.1007/978-0-8176-8364-1 |zbl=1277.05001 }}</ref> This is related to other appearances of the number seven in relation to [[exceptional object]]s, like the fact that the [[octonion]]s contain seven distinct square roots of −1, [[seven-dimensional cross product|seven-dimensional vectors]] have a [[cross product]], and the number of [[equiangular lines]] possible in seven-dimensional space is anomalously large.<ref>{{Cite journal |url=https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf |title=Cross products of vectors in higher dimensional Euclidean spaces |first=William S. |last=Massey |author-link=William S. Massey |journal=The American Mathematical Monthly |volume=90 |issue=10 |publisher=[[Taylor & Francis, Ltd]] |date=December 1983 |pages=697–701 |doi=10.2307/2323537 |jstor=2323537 |s2cid=43318100 |zbl=0532.55011 |access-date=2023-02-23 |archive-date=2021-02-26 |archive-url=https://web.archive.org/web/20210226011747/https://pdfs.semanticscholar.org/1f6b/ff1e992f60eb87b35c3ceed04272fb5cc298.pdf |url-status=dead }}</ref><ref>{{Cite journal |last1=Baez |first1=John C. |author-link=John Baez |url=http://math.ucr.edu/home/baez/octonions/ |title=The Octonions |journal=Bulletin of the American Mathematical Society |volume=39 |issue=2 |publisher=[[American Mathematical Society]] |pages=152–153 |year=2002 |doi=10.1090/S0273-0979-01-00934-X |mr=1886087|s2cid=586512 |doi-access=free }}</ref><ref>{{Cite book|last=Stacey |first=Blake C. |title=A First Course in the Sporadic SICs |date=2021 |publisher=Springer |isbn=978-3-030-76104-2 |location=Cham, Switzerland |pages=2–4 |oclc=1253477267}}</ref>[[File:Dice Distribution (bar).svg|thumb|Graph of the probability distribution of the sum of two six-sided dice]]

The lowest known dimension for an [[exotic sphere]] is the seventh dimension.<ref>{{Cite journal |last1=Behrens |first1=M. |last2=Hill |first2=M. |last3=Hopkins |first3=M. J. |last4=Mahowald |first4=M. |date=2020 |title=Detecting exotic spheres in low dimensions using coker J |url=https://onlinelibrary.wiley.com/doi/abs/10.1112/jlms.12301 |journal=Journal of the London Mathematical Society |publisher=[[London Mathematical Society]] |volume=101 |issue=3 |pages=1173 |arxiv=1708.06854 |doi=10.1112/jlms.12301 |mr=4111938 |s2cid=119170255 |zbl=1460.55017}}</ref><ref>{{Cite OEIS|A001676|Number of h-cobordism classes of smooth homotopy n-spheres.|access-date=2023-02-23}}</ref>

In [[hyperbolic space]], 7 is the highest dimension for non-simplex [[Coxeter–Dynkin diagram#Hypercompact Coxeter groups (Vinberg polytopes)|hypercompact ''Vinberg polytopes'']] of rank ''n + 4'' mirrors, where there is one unique figure with eleven [[Facet (geometry)|facets]]. On the other hand, such figures with rank ''n + 3'' mirrors exist in dimensions 4, 5, 6 and 8; ''not'' in 7.<ref>{{Cite journal |last1=Tumarkin |first1=Pavel |last2=Felikson |first2=Anna |url=https://www.ams.org/journals/mosc/2008-69-00/S0077-1554-08-00172-6/S0077-1554-08-00172-6.pdf |title=On ''d''-dimensional compact hyperbolic Coxeter polytopes with ''d + 4'' facets |journal=Transactions of the Moscow Mathematical Society |volume=69 |publisher=[[American Mathematical Society]] (Translation) |location=Providence, R.I. |year=2008 |pages=105–151 |doi= 10.1090/S0077-1554-08-00172-6 |doi-access=free |mr=2549446 |s2cid=37141102 |zbl=1208.52012 }}</ref>

There are seven fundamental types of [[catastrophe theory|catastrophes]].<ref>{{Cite book|last1=Antoni|first1=F. de|url=https://books.google.com/books?id=3L_sCAAAQBAJ&q=seven+fundamental+types+of+catastrophes&pg=PA13|title=COMPSTAT: Proceedings in Computational Statistics, 7th Symposium held in Rome 1986|last2=Lauro|first2=N.|last3=Rizzi|first3=A.|date=2012-12-06|publisher=Springer Science & Business Media|isbn=978-3-642-46890-2|pages=13|quote=...every catastrophe can be composed from the set of so called elementary catastrophes, which are of seven fundamental types.}}</ref>

When rolling two standard six-sided [[dice]], seven has a 1 in 6 probability of being rolled, the greatest of any number.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Dice|url=https://mathworld.wolfram.com/Dice.html|access-date=2020-08-25|website=mathworld.wolfram.com}}</ref> The opposite sides of a standard six-sided die always add to 7.

The [[Millennium Prize Problems]] are seven problems in [[mathematics]] that were stated by the [[Clay Mathematics Institute]] in 2000.<ref>{{Cite web |title=Millennium Problems {{!}} Clay Mathematics Institute |url=http://www.claymath.org/millennium-problems |access-date=2020-08-25 |website=www.claymath.org}}</ref> Currently, six of the problems remain [[unsolved problems in mathematics|unsolved]].<ref>{{Cite web |date=2013-12-15 |title=Poincaré Conjecture {{!}} Clay Mathematics Institute |url=http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-url=https://web.archive.org/web/20131215120130/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-date=2013-12-15 |access-date=2020-08-25}}</ref>

===Basic calculations===
{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[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
|-
|'''7 × ''x'''''
|'''7'''
|{{num|14}}
|{{num|21}}
|{{num|28}}
|{{num|35}}
|{{num|42}}
|{{num|49}}
|{{num|56}}
|{{num|63}}
|{{num|70}}
|{{num|77}}
|{{num|84}}
|{{num|91}}
|{{num|98}}
|{{num|105}}
|{{num|112}}
|{{num|119}}
|{{num|126}}
|{{num|133}}
|{{num|140}}
|{{num|147}}
|{{num|154}}
|{{num|161}}
|{{num|168}}
|{{num|175}}
|{{num|350}}
|{{num|700}}
|{{num|7000}}
|}

{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Division (mathematics)|Division]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
|-
|'''7 ÷ ''x'''''
|'''7'''
|3.5
|2.{{overline|3}}
|1.75
|1.4
|1.1{{overline|6}}
|rowspan=2 |[[1]]
|0.875
|0.{{overline|7}}
|0.7
|0.{{overline|63}}
|0.58{{overline|3}}
|0.{{overline|538461}}
|0.5
|0.4{{overline|6}}
|-
|'''''x'' ÷ 7'''
|0.<span style="text-decoration:overline">142857</span>
|0.<span style="text-decoration:overline">285714</span>
|0.<span style="text-decoration:overline">428571</span>
|0.<span style="text-decoration:overline">571428</span>
|0.<span style="text-decoration:overline">714285</span>
|0.<span style="text-decoration:overline">857142</span>
|1.<span style="text-decoration:overline">142857</span>
|1.<span style="text-decoration:overline">285714</span>
|1.<span style="text-decoration:overline">428571</span>
|1.<span style="text-decoration:overline">571428</span>
|1.<span style="text-decoration:overline">714285</span>
|1.<span style="text-decoration:overline">857142</span>
|{{num|2}}
|2.<span style="text-decoration:overline">142857</span>
|}

{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Exponentiation]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
|-
|'''7<sup>''x''</sup>'''
|'''7'''
|{{num|49}}
|{{num|343}}
|2401
|16807
|117649
|823543
|5764801
|40353607
|282475249
|1977326743
|13841287201
|96889010407
|-
|'''''x''<sup>7</sup>'''
|[[1]]
|{{num|128}}
|2187
|16384
|78125
|279936
|823543
|2097152
|4782969
|{{num|10000000}}
|19487171
|35831808 
|62748517
|}

====Decimal calculations====

{{num|999,999}} divided by 7 is exactly {{num|142,857}}. Therefore, when a [[vulgar fraction]] with 7 in the [[denominator]] is converted to a [[decimal]] expansion, the result has the same six-[[numerical digit|digit]] repeating sequence after the decimal point, but the sequence can start with any of those six digits.<ref>Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 82</ref> In [[decimal]] representation, the [[Multiplicative inverse|reciprocal]] of 7 repeats six [[Numerical digit|digits]] (as 0.{{overline|142857}}),<ref>{{Cite book |last=Wells |first=D. |url=https://archive.org/details/penguindictionar0000well_f3y1/mode/2up |title=The Penguin Dictionary of Curious and Interesting Numbers |publisher=[[Penguin Books]] |year=1987 |isbn=0-14-008029-5 |location=London |pages=171–174 |oclc=39262447 |url-access=registration |s2cid=118329153}}</ref><ref>{{Cite OEIS|A060283|Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's moved to end).|access-date=2024-04-02}}</ref> whose sum when [[Cyclic number#Relation to repeating decimals|cycling]] back to [[1]] is equal to 28.