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'''Büchi's problem''', also known as the '''''n'' squares' problem''', is an open problem from [[number theory]] named after the Swiss mathematician [[Julius Richard Büchi]]. It asks whether there is a positive integer ''M'' such that every sequence of ''M'' or more integer squares, whose second difference is constant and equal to 2, is necessarily a sequence of squares of the form (''x''&nbsp;+&nbsp;''i'')<sup>2</sup>, ''i''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;...,&nbsp;''M'',... for some integer&nbsp;''x''. In 1983, [[Douglas Hensley]] observed that Büchi's problem is equivalent to the following: Does there exist a positive integer ''M'' such that, for all integers ''x'' and ''a'', the quantity (''x''&nbsp;+&nbsp;''n'')<sup>2</sup>&nbsp;+&nbsp;''a'' cannot be a square for more than ''M'' consecutive values of&nbsp;''n'', unless&nbsp;''a''&nbsp;=&nbsp;0?
After another weekend of  [http://tinyurl.com/nqbkz6z uggs on sale] Six Nations activity the trophy is now a three horse race with Ireland, England and France all in contention with just one game left to play.<br><br>


==Statement of Büchi's problem==
In the final match however it is England who have the easier task by playing Italy whilst France take on Ireland, meaning the title could be decided on points scored, with Ireland currently in the [http://tinyurl.com/nqbkz6z ugg outlet] lead.<br><br>It has been an intensive few weeks as players are juggling club and country and pushing their bodies to ensure that are in fine form for each upcoming match. The constant wear  [http://tinyurl.com/nqbkz6z discount ugg boots] and tear of playing week in weekly out can takes its toll on the body, with some rugby injuries more common than others<br>There are more injuries in rugby per 1,000 hours of playing time than that of professional football, with 82 injuries compared to only 8. It is the high intensity nature of the sport which attracts spectators and players alike and rugby injuries is something that players manage as they happen.<br><br>The most common injuries in professional rugby between 2002 and 2012 were:<br>Thigh haematoma<br>A haematoma is an accumulation of blood which can result in swelling, pain and reduced function of the thigh muscle. There are a high percentage of these types of condition in rugby due to the contact nature of the sport with a players shoulder colliding with the thigh muscle during tackles.<br><br>In the immediate aftermath of diagnosis it is important to rest, use ice and compression to help manage any pain and inflammation as well as elevating the leg above the heart to reduce swelling.<br>Compression shorts can be used to manage the condition, working by applying pressure to the affected area and reducing muscle vibration during movement. Some products are available with a cinch strap whereby the patient is able to dictate the level of compression afforded to the thigh region.<br><br>Hamstring muscle injury (excluding haematomas)<br>The hamstring muscles are at their most active when an individual is running, sprinting or jumping with any strain to the muscles being quite painful and limiting movement of the player. There are three varying degrees of severity of injury with the grading dictating the treatment required and the overall recovery period.<br><br><br>A grade one tear is considered relatively minor and the symptoms should expect to subside following a period of rest, whilst a grade three tear could require surgery to repair the muscle followed by intensive physiotherapy to rebuild strength in the area. The latter can also result in heavy bruising and bleeding of the muscle.<br><br>As with thigh injuries compression shorts can be used to help manage hamstring injuries during your return to fitness. By applying pressure  [http://tinyurl.com/nqbkz6z ugg boots] to the hamstrings it can help to reduce the risk of further injury by preventing any unnatural movement of the muscle which can enhance athletic performance.<br><br>Concussion<br>This occurs when a blow to the head disrupts the brain and causes a short lived loss of [http://Www.Sharkbayte.com/keyword/mental+function mental function]. In rugby the risk of impact to the head during tackling and scrumming raises the risk of concussion. Concussions are not normally serious but need to be monitored as more severe conditions could occur.<br>Symptoms of concussion include headache, dizziness, sickness, loss of balance, confusion, issues with vision and in some cases memory loss. There are other symptoms which would require emergency medical treatment, including prolonged unconsciousness, difficulty staying awake and bleeding from the ears.<br><br>Please refer to the NHS website for full advice on concussion.<br>Knee - Medial Collateral Ligament injury<br>The knee joint is made up of four separate ligaments responsible for stabilising the joint in that of the Anterior Cruciate Ligament (ACL), Posterior Cruciate Ligament (PCL), Medial Collateral Ligament (MCL) and Lateral Collateral Ligament (LCL).<br><br>The MCL is common among the rugby injuries list on account of the type of tackles made which can force the knee inwards, with any sudden change if direction putting a strain on the ligament. A player should expect to recover from a mild strain within a few weeks  [http://tinyurl.com/nqbkz6z ugg boots sale] following a period of rest, ice to help manage any inflammation and even the use of a knee brace for additional stability of the joint.<br><br>Ankle lateral ligament injury<br>The main ligament within the ankle responsible for overall stability resides on the outside of the joint and damage here can prevent a player from being able to walk, let alone run. Each injury is completely different and whilst mild damage to the ligament can take a few days to recover from a tear of the ligament can require surgery to remedy.<br><br>Following any ankle injury strengthening the ligament is paramount to reducing the risk of further injury in the future, since the entire weight of the body passes through the joint when in motion. While rest is very important to your recovery an ankle brace can also be  [http://tinyurl.com/nqbkz6z discount ugg boots] worn to offer additional stability during movement.
 
Büchi's problem can be stated in the following way: Does there exist a positive integer ''M'' such that the system of equations
 
: <math>
\begin{cases}
x_2^2-2x_1^2+x_0^2=2\\
x_3^2-2x_2^2+x_1^2=2\\
{}\quad\vdots\\
x_{M-1}^2-2x_{M-2}^2+x_{M-3}^2=2
\end{cases}
</math>
 
has only solutions satisfying <math>x_n^2=(x_0+n)^2.</math>
 
Since the first difference of the sequence <math>\sigma=(x_n^2)_{n=0,\dots,M-1}</math> is the sequence <math>\Delta^{(1)}(\sigma)=(x_{n+1}^2-x_n^2)_{n=0,\dots,M-2}</math>, the second difference of <math>\sigma</math> is
 
: <math>
\Delta^{(2)}(\sigma)=((x_{n+2}^2-x_{n+1}^2)-(x_{n+1}^2-x_n^2))_{n=0,\dots,M-3}=(x_{n+2}^2-2x_{n+1}^2+x_n^2)_{n=0,\dots,M-3}.
</math>
 
Therefore, the above system of equations is equivalent to the single equation
 
: <math>\Delta^{(2)}(\sigma)=(2)_{n=0,\dots,M-3}</math>
 
where the unknown is the sequence <math>\sigma</math>.
 
==Examples==
 
Observe that for any integer ''x'' we have
 
: <math>(\star)\qquad(x+2)^2-2(x+1)^2+x^2=2. </math>
 
Hence the equation <math> x_2^2-2x_1^2+x_0^2=2</math> has solutions, called ''trivial Büchi sequences of length three'', such that <math>x_2^2=(x_0+2)^2</math> and <math>x_1^2=(x_0+1)^2</math>. For example, the sequences (2,&nbsp;3,&nbsp;4) and (2,&nbsp;&minus;3,&nbsp;4) are trivial Büchi sequences. A ''nontrivial Büchi sequence of length three'' is given for example by the sequence (0,&nbsp;7,&nbsp;10), as it satisfies 10<sup>2</sup>&nbsp;&minus;&nbsp;2·7<sup>2</sup>&nbsp;+&nbsp;0<sup>2</sup>&nbsp;=&nbsp;2, while 0<sup>2</sup>, 7<sup>2</sup> and 10<sup>2</sup> are not consecutive squares.
 
Replacing ''x'' by ''x''&nbsp;+&nbsp;1 in equation <math>(\star)</math>, we obtain <math>(x+3)^2-2(x+2)^2+(x+1)^2=2</math>. Hence the system of equations
 
: <math>
\begin{cases}
x_2^2-2x_1^2+x_0^2=2\\
x_3^2-2x_2^2+x_1^2=2
\end{cases}
</math>
 
has trivial Büchi solutions of length 4, namely the one satisfying <math>x_n^2=(x_0+n)^2</math> for ''n''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;3. In 1983, D. Hensley showed that there are infinitely many nontrivial Büchi sequences of length four. It is not known whether there exist any non-trivial Büchi sequence of length five (Indeed, Büchi asked originally the question only for&nbsp;''M''&nbsp;=&nbsp;5.).
 
==Original motivation==
A positive answer to Büchi's problem would imply, using the negative answer to [[Hilbert's Tenth Problem]] by [[Yuri Matiyasevich]], that there is no algorithm to [[Decidability (logic)|decide]] whether a system of diagonal [[quadratic form]]s with integer coefficients represents an integer tuple. Indeed, Büchi observed that squaring, therefore multiplication, would be existentially definable in the integers over the [[first-order logic|first-order]] language having two symbols of constant for 0 and 1, a symbol of function for the sum, and a symbol of relation ''P'' to express that an integer is a square.
 
==Some results==
 
[[Paul Vojta]] proved in 1999 that a positive answer to Büchi's Problem would follow from a positive answer to a weak version of the [[Bombieri–Lang conjecture]]. In the same article, he proves that the analogue of Büchi's Problem for the field of meromorphic functions over the complex numbers has a positive answer. Positive answers to analogues of Büchi's Problem in various other rings of functions have been obtained since then (in the case of rings of functions, one adds the hypothesis that not all ''x''<sub>''n''</sub> are constant).  
 
== References ==
*[[Paul Vojta|Vojta, Paul]] (1999), ''Diagonal quadratic forms and Hilbert’s tenth problem'', pp.&nbsp;261–274 in ''Hilbert’s tenth problem: relations with arithmetic and algebraic geometry'' (Ghent, 1999), edited by J. Denef et al., Contemp. Math. 270, Amer. Math. Soc., Providence, RI, 2000.
*[[Leonard Lipshitz|Lipshitz, Leonard]] (1990), "Quadratic forms, the five square problem, and diophantine equations" in ''Collected Papers of J. Richard Büchi''. Edited by Saunders Mac Lane and Dirk Siefkes. Springer, New York.
*[[Douglas Hensley|Hensley, Douglas]] (1983), “Sequences of squares with second difference of two and a conjecture of Büchi”, unpublished.
 
<!--- Categories --->
 
{{DEFAULTSORT:Buchi's Problem}}
[[Category:Number theory]]
[[Category:Quadratic forms]]

Latest revision as of 19:44, 29 November 2014

After another weekend of uggs on sale Six Nations activity the trophy is now a three horse race with Ireland, England and France all in contention with just one game left to play.

In the final match however it is England who have the easier task by playing Italy whilst France take on Ireland, meaning the title could be decided on points scored, with Ireland currently in the ugg outlet lead.

It has been an intensive few weeks as players are juggling club and country and pushing their bodies to ensure that are in fine form for each upcoming match. The constant wear discount ugg boots and tear of playing week in weekly out can takes its toll on the body, with some rugby injuries more common than others
There are more injuries in rugby per 1,000 hours of playing time than that of professional football, with 82 injuries compared to only 8. It is the high intensity nature of the sport which attracts spectators and players alike and rugby injuries is something that players manage as they happen.

The most common injuries in professional rugby between 2002 and 2012 were:
Thigh haematoma
A haematoma is an accumulation of blood which can result in swelling, pain and reduced function of the thigh muscle. There are a high percentage of these types of condition in rugby due to the contact nature of the sport with a players shoulder colliding with the thigh muscle during tackles.

In the immediate aftermath of diagnosis it is important to rest, use ice and compression to help manage any pain and inflammation as well as elevating the leg above the heart to reduce swelling.
Compression shorts can be used to manage the condition, working by applying pressure to the affected area and reducing muscle vibration during movement. Some products are available with a cinch strap whereby the patient is able to dictate the level of compression afforded to the thigh region.

Hamstring muscle injury (excluding haematomas)
The hamstring muscles are at their most active when an individual is running, sprinting or jumping with any strain to the muscles being quite painful and limiting movement of the player. There are three varying degrees of severity of injury with the grading dictating the treatment required and the overall recovery period.


A grade one tear is considered relatively minor and the symptoms should expect to subside following a period of rest, whilst a grade three tear could require surgery to repair the muscle followed by intensive physiotherapy to rebuild strength in the area. The latter can also result in heavy bruising and bleeding of the muscle.

As with thigh injuries compression shorts can be used to help manage hamstring injuries during your return to fitness. By applying pressure ugg boots to the hamstrings it can help to reduce the risk of further injury by preventing any unnatural movement of the muscle which can enhance athletic performance.

Concussion
This occurs when a blow to the head disrupts the brain and causes a short lived loss of mental function. In rugby the risk of impact to the head during tackling and scrumming raises the risk of concussion. Concussions are not normally serious but need to be monitored as more severe conditions could occur.
Symptoms of concussion include headache, dizziness, sickness, loss of balance, confusion, issues with vision and in some cases memory loss. There are other symptoms which would require emergency medical treatment, including prolonged unconsciousness, difficulty staying awake and bleeding from the ears.

Please refer to the NHS website for full advice on concussion.
Knee - Medial Collateral Ligament injury
The knee joint is made up of four separate ligaments responsible for stabilising the joint in that of the Anterior Cruciate Ligament (ACL), Posterior Cruciate Ligament (PCL), Medial Collateral Ligament (MCL) and Lateral Collateral Ligament (LCL).

The MCL is common among the rugby injuries list on account of the type of tackles made which can force the knee inwards, with any sudden change if direction putting a strain on the ligament. A player should expect to recover from a mild strain within a few weeks ugg boots sale following a period of rest, ice to help manage any inflammation and even the use of a knee brace for additional stability of the joint.

Ankle lateral ligament injury
The main ligament within the ankle responsible for overall stability resides on the outside of the joint and damage here can prevent a player from being able to walk, let alone run. Each injury is completely different and whilst mild damage to the ligament can take a few days to recover from a tear of the ligament can require surgery to remedy.

Following any ankle injury strengthening the ligament is paramount to reducing the risk of further injury in the future, since the entire weight of the body passes through the joint when in motion. While rest is very important to your recovery an ankle brace can also be discount ugg boots worn to offer additional stability during movement.

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