{"id":23559,"date":"2013-07-25T09:27:02","date_gmt":"2013-07-25T13:27:02","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=23559"},"modified":"2014-06-27T12:11:08","modified_gmt":"2014-06-27T17:11:08","slug":"odd-perfect-numbers-exist","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/07\/25\/odd-perfect-numbers-exist\/","title":{"rendered":"Odd Perfect Numbers: Do They Exist?"},"content":{"rendered":"

\"images-2\"<\/a>Mathematical inquiry can often lead to a jungle of unique questions and problems.\u00a0 In the field of Number Theory, there are a wide assortment of such mathematical creatures.\u00a0 Although these problems are easy to state, they can remain dormant for years with little sign of progress. \u00a0 In fact, the Odd Perfect Number Conjecture is one such problem that has escaped proof for centuries.<\/p>\n

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Perfect numbers are positive integers that are the sum of their proper divisors.\u00a0 For instance, 6 is a perfect number, because the sum of its proper divisors, 1, 2, and 3 equals 6 (1 + 2 + 3 = 6).\u00a0 Euclid first devised a way to construct a set of even perfect numbers in Book IX of\u00a0The Elements.\u00a0\u00a0<\/i>In his book, Euclid showed that if \"2^p- is prime, when \"p\" is prime, then \"2^ is a perfect number.\u00a0 From my last post on \u201cThe Infinitude of Mersenne Primes\u201d, one may recognize that if \"p\" and \u00a0\"2^p-1\" are prime, \u00a0then \"2^p-1\" is a Mersenne Prime.<\/p>\n

In 1638,\u00a0Ren\u00e9 Descartes\u00a0sent a letter to Marin Mersenne stating that he believed every even perfect number is of Euclid\u2019s form.\u00a0 Furthermore, in the letter, Descartes was the first to reason that an\u00a0odd perfect number\u00a0may or may not exist.\u00a0\u00a0 Many mathematicians since have failed to produce a proof.\u00a0 So, does there exist an odd perfect number?<\/p>\n

Computationally the conjecture has been checked for odd numbers up to \"10^{300}\" \u00a0with no success.\u00a0\u00a0\u00a0 Over time, mathematicians have produced several remarkable results.\u00a0 In 1888,\u00a0Eug\u00e8ne Charles Catalan proved that if an odd perfect number does exist and it is not divisible by 3, 5, or 7, then it has at least 26 prime factors (this result was later extended to 27 prime factors by K.K. Norton in 1960).\u00a0 Another remarkable result came from the mathematician J. Touchard.\u00a0\u00a0\u00a0 In 1953, Touchard showed that if an odd perfect number exists it must be of the form \"12k+1\" or \"36k.<\/p>\n

Resources and more examples can be found easily on the internet. \u00a0The Norwegian mathematician\u00a0\u00d8ystein Ore had the following to say about the conjecture and Euclid’s form in his book Invitation to Number Theory<\/i>:<\/p>\n

“This result shows that each Mersenne prime gives rise to a perfect number…. Are there any other types of perfect numbers?… This leaves us with the question: ARE THERE ANY ODD PERFECT NUMBERS? \u00a0Presently we know of none and it is one of the outstanding puzzles of number theory to determine \u00a0whether an odd perfect number can exist….”<\/em><\/p>\n

From Ore’s words, the conjecture is definitely an outstanding puzzle. \u00a0Elegance is a word that mathematicians use when describing a result that is parsimonious and rigorous. \u00a0It would be nice to see an elegant solution to this old conundrum. \u00a0One that exhibits robustness and breeds more questions of like interest and \u00a0uniqueness.<\/p>\n

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Mathematical inquiry can often lead to a jungle of unique questions and problems.\u00a0 In the field of Number Theory, there are a wide assortment of such mathematical creatures.\u00a0 Although these problems are easy to state, they can remain dormant for … Continue reading →<\/span><\/a><\/p>\n

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