{"id":23706,"date":"2013-08-08T09:25:18","date_gmt":"2013-08-08T13:25:18","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=23706"},"modified":"2014-06-26T16:25:54","modified_gmt":"2014-06-26T21:25:54","slug":"problem-brocard","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/08\/08\/problem-brocard\/","title":{"rendered":"The Problem Of Brocard"},"content":{"rendered":"

\"The<\/a>

The first page of Brocard’s original 1897 Paper. Photo Credit [ Wikimedia Commons]<\/p><\/div>Down through the river of time, rapids of mathematical imagination have emerged curving the path of linear thought.\u00a0\u00a0 Often times, problems arising from these rocky waters branch off into smaller streams and drift indefinitely avoiding the trappings of resolution.\u00a0 One such problem was introduced by the French mathematician Henri Brocard in 1876 and later, in a separate paper, in 1885.\u00a0 Brocard inquired about a set of possible positive\u00a0integers \"n\" such that the equation \"n! is satisfied.<\/p>\n

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The \"n!\" term in the equation is known as the factorial of \"n\".\u00a0 Basically, for any positive integer \"n\", \"n\" factorial is defined as the multiplicative product of all of the integers from 1 to \"n\".\u00a0 \u00a0For example, 4 factorial\u00a0is simply\u00a04\u00a0x 3\u00a0x 2\u00a0x 1 = 24.\u00a0 With this definition it is to see the obscurity surrounding the problem of finding a set of solutions that satisfy the equation. \u00a0The basic question still remains: For what positive integers \"n\" is the equation\u00a0\u00a0\"n! \u00a0satisfied for some positive integer \"m\"?<\/p>\n

There are only three known solutions.\u00a0 Ordered pairs (\"n\",\"m\") that satisfy the equation are known as Brown Numbers.\u00a0 The only three known pairs of Brown Numbers are (4, 5), (5, 11), and (7, 71).\u00a0 More formally, Brocard\u2019s Problem asks the question: Does $n$ and $m$ exist such that \"n! other than for \"n\" = 4, 5, and 7?\u00a0 In other words, Do more pairs of Brown Numbers exist other than (4,5), (5,11), and (7,71)?<\/p>\n

The prolific mathematician Paul Erd\u0151s conjectured that no other pairs of Brown Numbers exist.\u00a0 In 2000, mathematicians Bruce C. Berndt and William F. Galway published a paper, \u201cThe Brocard-Ramanujan diophantine equation \"n!\u201d in The Ramanujan Journal <\/i>\u00a0showing computational verification that no other solutions exist for \"n\" up to \"10^{9}\".\u00a0 Furthermore, it was proved by mathematicians A. Dabrowski (1996) and Florian Luca (2002), that the equation only has finitely many solutions assuming another major unsolved problem in Number Theory, the abc conjecture, is true.<\/p>\n

So, does there exist any more pairs of Brown Numbers?\u00a0 With the computational verification of Berndt and Galway one is tempted to say no.\u00a0 However, past mathematical inquiry has shown that computational verification does not always mean proof.\u00a0 \u00a0For some conjectures, counterexamples have been shown to creep up just beyond extensive computational rendering.\u00a0 In any case, Brocard\u2019s Problem is still gushing in the rapids of mathematical imagination.<\/p>\n

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Down through the river of time, rapids of mathematical imagination have emerged curving the path of linear thought.\u00a0\u00a0 Often times, problems arising from these rocky waters branch off into smaller streams and drift indefinitely avoiding the trappings of resolution.\u00a0 One … Continue reading →<\/span><\/a><\/p>\n

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