{"id":25131,"date":"2014-10-14T22:06:00","date_gmt":"2014-10-15T03:06:00","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=25131"},"modified":"2014-10-14T22:09:08","modified_gmt":"2014-10-15T03:09:08","slug":"carmichaels-totient-conjecture","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2014\/10\/14\/carmichaels-totient-conjecture\/","title":{"rendered":"Carmichael’s Totient Conjecture"},"content":{"rendered":"
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Photo Credit: Avery Carr<\/p><\/div>\n

In the wake of mathematical enlightenment a profound understanding of basic notions bridges the gap between the conceptual and concrete.\u00a0\u00a0 In many cases, problems that have an exterior of simplicity exploit the boundaries of comprehension and provide insight into extensive associations.\u00a0 From the mind-stretching inclinations of geometry and algebra emerges the intricate framework from which these connections form.\u00a0 Piece by piece, generalizations\u00a0are built from the material of empirical understanding fabricated by the process of asking intrinsic questions.<\/p>\n

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Questions of this nature are\u00a0entwined in reticent patterns found across the full spectrum of mathematics. \u00a0 Many\u00a0of these inquiries encompass and ascertain the properties of special functions. \u00a0 \u00a0 Mappings such as Euler’s Totient function provide a strong basis for further investigation into\u00a0characteristics of positive integers. \u00a0 Specifically, \u00a0this function denoted by \"\phi\u00a0 counts the number of positive integers less than or equal to a positive integer \"n\" such that the positive integers counted and \"n\" have only 1 as their common divisor (in other words they are relatively prime to \"n\", denoted \"\gcd(x,n) \u00a0such that \u00a0\"x).\u00a0 \u00a0In example,\u00a0 \"\phi = \u00a0\"2\" \u00a0because there are two positive integers less than or equal to 3 that are relatively prime to 3 (namely 1 and 2, given \"gcd(3,2)=gcd(3,1)). Euler\u2019s Totient function is distinguished\u00a0by several other properties as well.\u00a0\u00a0\u00a0 For instance, if \"p\" is a prime number, then \"\phi(p).\u00a0\u00a0 It is also multiplicative, in the sense that if \"\ \u00a0then \u00a0\"\phi = \"\phi. \u00a0 By virtue of these attributes, several open problems in the field of number theory involve \"\phi.<\/p>\n

The mathematician Robert Carmichael proposed one such conundrum\u00a0in 1907 that still remains\u00a0unsolved. \u00a0\u00a0Basically, \u00a0Carmichael conjectured that for every positive integer \"n\" there exists a positive integer \"m\" such that \u00a0\"m and \"\phi. \u00a0 As a consequence, with the given properties, the conjecture is certainly true for odd numbers. \u00a0This can be seen by letting \"n\" be a positive odd integer and in the fact that\u00a0\"\phi =\u00a0<\/span>\u00a0\"1\", which it follows \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/p>\n

\"\phi = \"\phi = \"\phi.<\/p>\n

However, \u00a0as easily proved as the conjecture is for the positive odd integers, \u00a0the statement has not been shown true for the positive even integers. \u00a0 Maybe a clever argument will come from a thorough investigation of basic notions. \u00a0\u00a0 Perhaps, rather, it will be stumbled upon in search of greater abstractions. \u00a0Whatever the case of discovery may be, a resolution will certainly be achieved, if at all, by asking insightful\u00a0questions.<\/p>\n

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In the wake of mathematical enlightenment a profound understanding of basic notions bridges the gap between the conceptual and concrete.\u00a0\u00a0 In many cases, problems that have an exterior of simplicity exploit the boundaries of comprehension and provide insight into extensive … Continue reading →<\/span><\/a><\/p>\n

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