{"id":24033,"date":"2013-09-18T20:34:26","date_gmt":"2013-09-19T00:34:26","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=24033"},"modified":"2014-06-24T13:34:52","modified_gmt":"2014-06-24T18:34:52","slug":"equivalent-statements-strong-goldbach-conjecture","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/09\/18\/equivalent-statements-strong-goldbach-conjecture\/","title":{"rendered":"The Strong Goldbach Conjecture: An Equivalent Statement"},"content":{"rendered":"

\"phi\"<\/a>Some problems in mathematics remain buried deep in the catacombs of slow progress.\u00a0\u00a0 Whether in Algebra, Topology, Discrete Math, or Analysis, mind-stretching mysteries await to be discovered beyond the boundaries of former thought.\u00a0 Frequently, solutions to such problems come from a clever combination of previously known results.\u00a0\u00a0 In conjunction with this concept, it is often helpful to reformulate open problems into equivalent statements to gain different perspectives.<\/p>\n

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As an example, the unsolved Strong Goldbach Conjecture, that proposes every even integer greater than 2 is the sum of two prime numbers, can be reformulated into an equation involving Euler\u2019s Totient function.\u00a0\u00a0 By definition, two integers \"m\" and \"n\" are said to be relatively prime if the greatest common divisor of \"m\" and \"n\" is 1, denoted \"gcd(m,n).\u00a0 For instance, let \"m\" = 3 and \"n\" = 5.\u00a0 Since there are no divisors greater than 1 that is shared between 3 and 5, notably both being prime numbers, the \"gcd(3,5).\u00a0 Therefore, 3 and 5 are said to be relatively prime.\u00a0\u00a0 With this in mind, Euler\u2019s Totient function, denoted \"\phi, is the total number of positive integers less than or equal to \"n\" such that they are relatively prime to \"n\".<\/p>\n

Intuitively, for every prime number p and integer \"k, \"gcd(p,k) , and therefore \"\phi\u00a0 = \"p\" \u00a0\u2013 1.\u00a0\u00a0\u00a0 Given this, the statement of the Strong Goldbach Conjecture can be restated as: for every integer \"n, there\u00a0exists primes p and q such that \"\phi(p) The mathematicians Paul Erd\u0151s and Leo Moser further asked whether or not there exists any integers p and q, not necessarily prime, such that the reformulated statement holds.<\/p>\n

In considering the proposition of\u00a0Erd\u0151s\u00a0\u00a0and Moser, such questions provide a good starting point for solving the harder problem of considering p and q such that they are necessarily prime. \u00a0 It does so by seeking to expose the nature of the equation in general. \u00a0Reformulations of this type provide a digging tool that could one day unearth a proof to the Strong Goldbach Conjecture, and other unsolved conundrums, from the catacombs of mathematical lethargy.<\/p>\n

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Some problems in mathematics remain buried deep in the catacombs of slow progress.\u00a0\u00a0 Whether in Algebra, Topology, Discrete Math, or Analysis, mind-stretching mysteries await to be discovered beyond the boundaries of former thought.\u00a0 Frequently, solutions to such problems come from … Continue reading →<\/span><\/a><\/p>\n

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