61 (number): Difference between revisions
Jump to navigation
Jump to search
en>AnomieBOT m Dating maintenance tags: {{Citation needed}} |
en>David Eppstein →In mathematics: remove unsourced fact not in OEIS |
||
| Line 1: | Line 1: | ||
{{Infobox number | |||
| number = 68 | |||
| divisor = 1, 2, 4, 17, 34, 68 | |||
}} | |||
'''68''' ('''sixty-eight''') is the [[natural number]] following [[67 (number)|67]] and preceding [[69 (number)|69]]. | |||
==In mathematics== | |||
68 is a [[Perrin number]].<ref>{{SloanesRef|sequencenumber=A001608|name=Perrin sequence (or Ondrej Such sequence): a(n) = a(n-2) + a(n-3)}}</ref> | |||
It is the largest known number to be the sum of two primes in exactly two different ways: 68 = 7 + 61 = 31 + 37.<ref>http://math.fau.edu/richman/Interesting/WebSite/Number68.pdf retrieved 13 March 2013</ref> All higher even numbers that have been checked are the sum of three or more pairs of primes; the conjecture that 68 is the largest number with this property is closely related to the [[Goldbach conjecture]] and like it remains unproven.<ref>{{SloanesRef|sequencenumber=A000954|name=Conjecturally largest even integer which is an unordered sum of two primes in exactly n ways}}</ref> | |||
Because of the factorization of 68 as <math>2^2\cdot (2^{2^2}+1)</math>, a 68-sided [[regular polygon]] may be constructed with [[compass and straightedge]].<ref>{{SloanesRef|sequencenumber=A003401|name=Numbers of edges of polygons constructible with ruler and compass}}</ref> | |||
[[File:Tamari lattice.svg|thumb|A Tamari lattice, with 68 upward paths of length zero or more from one element of the lattice to another.]] | |||
There are exactly 68 10-bit [[binary number]]s in which each bit has an adjacent bit with the same value,<ref>{{SloanesRef|sequencenumber=A006355|name=Number of binary vectors of length n containing no singletons}}</ref> exactly 68 combinatorially distinct [[Point set triangulation|triangulations]] of a given triangle with four points interior to it,<ref name="A000260">{{SloanesRef|sequencenumber=A000260|name=Number of rooted simplicial 3-polytopes with n+3 nodes}}</ref>and exactly 68 [[Partially ordered set#Interval|intervals]] in the [[Tamari lattice]] describing the ways of parenthesizing five items.<ref name="A000260"/> The largest [[graceful graph]] on 13 nodes has exactly 68 edges.<ref>{{SloanesRef|sequencenumber=A004137|name=Maximal number of edges in a graceful graph on n nodes}}</ref> There are 68 different [[undirected graph]]s with six edges and no isolated nodes,<ref>{{SloanesRef|sequencenumber=A000664|name=Number of graphs with n edges}}</ref> 68 different minimally [[k-vertex-connected graph|2-connected graphs]] on seven unlabeled nodes,<ref>{{SloanesRef|sequencenumber=A003317|name=Number of unlabeled minimally 2-connected graphs with n nodes (also called "blocks")}}</ref> 68 different [[degree sequence]]s of four-node connected graphs,<ref>{{SloanesRef|sequencenumber=A007721|name=Number of distinct degree sequences among all connected graphs with n nodes}}</ref> and 68 [[matroid]]s on four labeled elements.<ref>{{SloanesRef|sequencenumber=A058673|name=Number of matroids on n labeled points}}</ref> | |||
[[Størmer's theorem]] proves that, for every number ''p'', there are a finite number of pairs of consecutive numbers that are both [[smooth number|''p''-smooth]] (having no prime factor larger than ''p''). For ''p'' = 13 this finite number is exactly 68.<ref>{{SloanesRef|sequencenumber=A002071|name=Number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most the n-th prime}}</ref> On an infinite chessboard, there are 68 squares three knight's moves away from any cell.<ref>{{SloanesRef|sequencenumber=A018842|name=Number of squares on infinite chess-board at n knight's moves from center}}</ref> | |||
As a [[decimal number]], 68 is the last two-digit number to appear in the digits of [[pi]].<ref>{{SloanesRef|sequencenumber=A032510|name=Scan decimal expansion of Pi until all n-digit strings have been seen; a(n) is last string seen}}</ref> It is a [[happy number]], meaning that repeatedly summing the squares of its digits eventually leads to 1:<ref>{{SloanesRef|sequencenumber=A007770|name=Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map includes 1}}</ref> | |||
:<math>68\to 6^2+8^2=100\to 1^2+0^2+0^2=1.</math> | |||
==Other uses== | |||
*68 is the [[atomic number]] of [[erbium]], a lanthanide | |||
*In the restaurant industry, 68 may be used as a code meaning "put back on the menu", being the opposite of [[86 (number)|86]] which means "remove from the menu".<ref>{{citation|title=Words at Work: An Insider’s Guide to the Language of Professions|first=Mim|last=Harrison|publisher=Bloomsbury Publishing USA|year=2009|isbn=9780802718686|page=7|url=http://books.google.com/books?id=dAf_HxF1HTwC&pg=PA7}}.</ref> | |||
*68 may also be used as [[slang]] for [[oral sex]], based on a play on words involving the number [[69 (number)|69]].<ref>{{citation|title=The Concise New Partridge Dictionary of Slang and Unconventional English|first1=Terry|last1=Victor|first2=Tom|last2=Dalzell|edition=8th|publisher=Psychology Press|year=2007|isbn=9780203962114|page=585|url=http://books.google.com/books?id=7UIjVGcSe8MC&pg=PA585}}.</ref> | |||
==See also== | |||
*[[68 (disambiguation)]] | |||
==References== | |||
{{Reflist}} | |||
{{Integers|zero}} | |||
[[Category:Integers]] | |||
Revision as of 08:54, 31 October 2013
Template:Infobox number 68 (sixty-eight) is the natural number following 67 and preceding 69.
In mathematics
68 is a Perrin number.[1]
It is the largest known number to be the sum of two primes in exactly two different ways: 68 = 7 + 61 = 31 + 37.[2] All higher even numbers that have been checked are the sum of three or more pairs of primes; the conjecture that 68 is the largest number with this property is closely related to the Goldbach conjecture and like it remains unproven.[3]
Because of the factorization of 68 as , a 68-sided regular polygon may be constructed with compass and straightedge.[4]
There are exactly 68 10-bit binary numbers in which each bit has an adjacent bit with the same value,[5] exactly 68 combinatorially distinct triangulations of a given triangle with four points interior to it,[6]and exactly 68 intervals in the Tamari lattice describing the ways of parenthesizing five items.[6] The largest graceful graph on 13 nodes has exactly 68 edges.[7] There are 68 different undirected graphs with six edges and no isolated nodes,[8] 68 different minimally 2-connected graphs on seven unlabeled nodes,[9] 68 different degree sequences of four-node connected graphs,[10] and 68 matroids on four labeled elements.[11]
Størmer's theorem proves that, for every number p, there are a finite number of pairs of consecutive numbers that are both p-smooth (having no prime factor larger than p). For p = 13 this finite number is exactly 68.[12] On an infinite chessboard, there are 68 squares three knight's moves away from any cell.[13]
As a decimal number, 68 is the last two-digit number to appear in the digits of pi.[14] It is a happy number, meaning that repeatedly summing the squares of its digits eventually leads to 1:[15]
Other uses
- 68 is the atomic number of erbium, a lanthanide
- In the restaurant industry, 68 may be used as a code meaning "put back on the menu", being the opposite of 86 which means "remove from the menu".[16]
- 68 may also be used as slang for oral sex, based on a play on words involving the number 69.[17]
See also
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ Template:SloanesRef
- ↑ http://math.fau.edu/richman/Interesting/WebSite/Number68.pdf retrieved 13 March 2013
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ 6.0 6.1 Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Template:SloanesRef
- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.