{"id":24161,"date":"2013-10-05T12:22:53","date_gmt":"2013-10-05T16:22:53","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=24161"},"modified":"2014-06-21T11:32:03","modified_gmt":"2014-06-21T16:32:03","slug":"solitary-10","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/10\/05\/solitary-10\/","title":{"rendered":"Solitary 10"},"content":{"rendered":"

\"images-1\"<\/a>Connections between integers are deeply studied in the field of Number Theory.\u00a0 In certain instances, these abstractions are attributed character that takes shape in tangible analogs. When referring to colleague and prodigy, Srinivasa Ramanujan, the mathematician John E. Littlewood remarked, \u201cEvery positive integer was one of his personal friends.\u201d<\/p>\n

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While these words speak of the uncanny ability Ramanujan had in seeing patterns and connections between numbers, the concept of friendship amongst numbers is a topic seriously considered by specialists in mathematics.\u00a0 In particular, number theorists have created a rule that defines whether any two natural numbers are friendly.\u00a0 Of course, friendship here is not the same as the classical understanding derived from the social interaction between humans.\u00a0 Moreover, it is simply a rule carried out on two natural numbers to evaluate if certain conditions are met. \u00a0If any two natural numbers meet these conditions, they are deemed a friendly pair.<\/p>\n

Rules in mathematics often involve the concept of a function.\u00a0 Functions can be loosely thought of as a machine that receives an input and produces an output.\u00a0\u00a0 If an object is placed in the input, either the same object or a different object comes out of the output.\u00a0 However, it is not considered a function if two different objects can come out of the output for one object in the input.<\/p>\n

The divisor function is a function that takes a natural number n and sums the divisors of n, denoted \"\sigma(n)\".\u00a0\u00a0 For example, {1,2,4} are the divisors of the natural number 4, because each of 1, 2, and 4 divide 4.\u00a0 Therefore , \"\sigma(4).\u00a0\u00a0 Let\u2019s further define the divisor function divided by n, \"\sigma(n)\/n\", known as the abundancy function denoted by \"\sum.<\/p>\n

With these definitions in hand, two natural numbers \"m\" and \"n\" are said to be a friendly pair, (\"m\",\"n\"), \u00a0if \"\sum(n)\" = \"\sum.\u00a0 Many pairs of numbers are known to be friendly pairs.\u00a0\u00a0 However, there are a few, with 10 being the least, in which there is no other known natural number \"m\" such that 10 and \"m\" are a friendly pair.\u00a0\u00a0 So, one can ask: Is 10 a solitary number?\u00a0 Of course, solitary here means there does not exist a natural number \"m\" in which \"m\" and 10 are a friendly pair.<\/p>\n

Showing that a number is not solitary can involve some rather large numbers.\u00a0 For instance, 24 is not a solitary number, because 24 and the number 91,963,648 are a friendly pair.\u00a0 Determining whether or not 10 is a solitary number is of great interest to number theorists and math hobbyists alike. Elegant approaches are yet to be found.\u00a0 However, the solution may come from a novel use of known connections etched permanently in the history of mathematics.<\/p>\n

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Connections between integers are deeply studied in the field of Number Theory.\u00a0 In certain instances, these abstractions are attributed character that takes shape in tangible analogs. When referring to colleague and prodigy, Srinivasa Ramanujan, the mathematician John E. Littlewood remarked, … Continue reading →<\/span><\/a><\/p>\n

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