{"id":24518,"date":"2014-02-15T22:41:06","date_gmt":"2014-02-16T03:41:06","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=24518"},"modified":"2014-06-18T14:59:22","modified_gmt":"2014-06-18T19:59:22","slug":"sierpinski-number-bust","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2014\/02\/15\/sierpinski-number-bust\/","title":{"rendered":"Sierpi\u0144ski Or Bust"},"content":{"rendered":"
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Photo Credit: Avery Carr<\/p><\/div>\n

The legendary math Universalist, Henri Poincare, once said, \u201c Mathematicians do not study objects, but relations between objects.\u201d\u00a0 Inspired by subtle patterns that emanate from an abyss of ostensible chaos, mathematicians manifest these words by exploring relations within algebraic and geometric structures.\u00a0 These edifices, upon occasion, reveal intricate beauty buried deep in a profusion of symbols etched on the tablet of logical thought. Problems requiring various degrees of acumen have been presented in an assortment of publications over the course of modern history.\u00a0 \u00a0From the field of Number Theory, problems are often innocently conveyed, but cloaked behind a veil of extensive logic that guards them from any immediate proof.\u00a0 \u00a0Some problems call for a deep understanding harnessed only after years of study. \u00a0 However, others are immediately accessible with a little mathematical maturity.\u00a0\u00a0 One such conundrum can be stated in terms of Sierpi\u0144ski\u00a0numbers of the second kind.\u00a0\u00a0 A Sierpi\u0144ski\u00a0number of the second kind (henceforth, just\u00a0Sierpi\u0144ski\u00a0number)\u00a0is an odd positive integer \"k\" such that \"k2^{n}+1\" is a composite number for every positive integer \"n\". In 1960, Wac\u0142aw\u00a0Sierpi\u0144ski\u00a0proved that there are infinitely many such \"k\".\u00a0\u00a0\u00a0 To eliminate a given positive integer \"k\" from Sierpi\u0144ski\u00a0number candidacy, one only needs to show a prime number of the form \"k2^n+1\" for some positive integer \"n\".\u00a0 On the other hand, to prove that \"k\" is a Sierpi\u0144ski\u00a0number, the convention is to show that there exists a covering set of primes such that every member of the set divides \"k2^n+1\" for every positive integer \"n\". The smallest Sierpi\u0144ski\u00a0number currently known is 78557 and was proven by the mathematician John Selfridge in 1962 with the covering set { 3, 5, 7, 13, 19, 37, 73 }. Therefore, a fundamental open question arises:\u00a0 Out of the set of all Sierpi\u0144ski\u00a0numbers, is 78557 the smallest?\u00a0 To answer this question in the affirmative, one has to eliminate all positive integers less than 78557 from Sierpi\u0144ski\u00a0number candidacy.\u00a0 Mathematical proof and computational methods have eliminated all but six remaining candidates: 10223, 21181, 22699, 24737, 55459, and 67607. The website Seventeen or Bust<\/a><\/em> is a distributed computing project dedicated to the goal of eliminating these remaining six.\u00a0\u00a0\u00a0 When the project first started there were seventeen\u00a0 (hence the name), \u00a0but since then, \u00a0eleven have been eliminated by finding the desired prime.\u00a0 Anyone can join the search. \u00a0 \u00a0 It could be you that proves that 78557 is the smallest Sierpi\u0144ski number. \u00a0 All one needs is a computer and a willingness to give up a small amount of computational power. \u00a0So plug in and discover.<\/p>\n

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The legendary math Universalist, Henri Poincare, once said, \u201c Mathematicians do not study objects, but relations between objects.\u201d\u00a0 Inspired by subtle patterns that emanate from an abyss of ostensible chaos, mathematicians manifest these words by exploring relations within algebraic and … Continue reading →<\/span><\/a><\/p>\n

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