{"id":23931,"date":"2013-08-22T11:29:20","date_gmt":"2013-08-22T15:29:20","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=23931"},"modified":"2014-06-24T13:50:05","modified_gmt":"2014-06-24T18:50:05","slug":"lonely-runner-conjecture","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/08\/22\/lonely-runner-conjecture\/","title":{"rendered":"The Lonely Runner Conjecture"},"content":{"rendered":"

\"Lonely<\/a>The eminent mathematician Carl Friedrich Gauss once said, \u201cMathematics is the queen of the sciences.\u201d\u00a0\u00a0\u00a0Considering this statement to be true, it is easy to see the span of her kingdom.\u00a0 From the design of airplane wings to the ever increasing speed of computation, the royal seal of mathematics is a permanent hallmark of industry and science.\u00a0\u00a0\u00a0 Her practitioners, both pure and applied, have pushed the boundaries of current thought into the realm of new abstractions.<\/p>\n

<\/p>\n

Sometimes these new areas are recreational in nature.\u00a0 In 1967, the mathematician J\u00f6rg M. Wills proposed a conjecture that fully embodies such novelty.\u00a0 The problem was later named by Luis Goddyn in 1998 as The Lonely Runner Conjecture.\u00a0 The formal statement of the conjecture is the following:<\/p>\n

The Lonely Runner Conjecture (1967):<\/b>\u00a0Suppose \"k\" runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1.\u00a0 Then, for any given runner, there is a point in time at which each runner is a distance of at least \"1\/k\" along the track away from every other runner.<\/p>\n

In pursuit of a counterexample or general proof, mathematicians have also explored specific values of \"k\".\u00a0 As of 2008,\u00a0the conjecture has been proved for every value of \"k\" up to 7. \u00a0 Interesting enough, the problem can be reformulated to the following question: Can the conjecture be proved for \"k?\u00a0 Reformulations of this character are particularly useful to researchers as they look for new perspectives and angles leading to an ultimate proof.<\/p>\n

However, a final proof could remain mummified in the tomb of abstract mathematics for centuries.\u00a0\u00a0 Mathematicians normally employ one of three methods of proof in their tomb raiding; mathematical induction, direct deduction, or contradiction.\u00a0 \u00a0\u00a0Of the three, it seems that mathematical induction or contradiction are the methods that will likely yield success for this particular problem.<\/p>\n

Mathematical induction is a method that seeks to establish a statement true for the integers by evaluating a base step for a variable, say \"n\", and then proceeding to show the statement is true for \"n in the inductive step.\u00a0 \u00a0The method of contradiction assumes a particular statement is true or false and proceeds to show a contradiction to the premise.\u00a0\u00a0\u00a0 For The Lonely Runner Conjecture one could imagine how these two methods are used.<\/p>\n

In the case of mathematical induction, one could possibly construct an algebraic reformulation of the conjecture that depends on the value of \"k\", the number of runners.\u00a0 With this, the base step of \"k runners could be used and is already shown to be true.\u00a0\u00a0 Assuming that the statement would be true for \"k\", the hard part would be to show it true for \"k.\u00a0 Alternatively, one could use contradiction by assuming a counterexample exists for some \"k\" as a premise, and then show this premise contradicts fundamental logic.<\/p>\n

These are just sketches of how one could approach a proof.\u00a0\u00a0\u00a0 The conjecture has escaped all attempts for more than 46 years.\u00a0 How would you try to prove it?\u00a0 Would you use induction or contradiction?<\/p>\n

It is not clear what will yield a final or partial answer.\u00a0\u00a0\u00a0 The solution could emerge from the tomb this year or remain shrouded for centuries.\u00a0 However, from the current state of industry and technology to a lonely runner sprinting on a track, the kingdom of mathematics is alive and well.<\/p>\n

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The eminent mathematician Carl Friedrich Gauss once said, \u201cMathematics is the queen of the sciences.\u201d\u00a0\u00a0\u00a0Considering this statement to be true, it is easy to see the span of her kingdom.\u00a0 From the design of airplane wings to the ever increasing … Continue reading →<\/span><\/a><\/p>\n

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