{"id":23360,"date":"2013-06-20T10:00:28","date_gmt":"2013-06-20T14:00:28","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=23360"},"modified":"2014-07-02T10:04:35","modified_gmt":"2014-07-02T15:04:35","slug":"conjecture-marie-sophie-germain","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/06\/20\/conjecture-marie-sophie-germain\/","title":{"rendered":"The Conjecture of Marie-Sophie Germain"},"content":{"rendered":"

\"images-1\"<\/a>Numbers pervade our lives in many different venues.\u00a0 Prime numbers, in particular, weave their way into the very fabric of our daily existence.\u00a0\u00a0 From surfing the internet to pseudo-random number generators, primes are found ever present behind a multitude of abstractions.\u00a0\u00a0\u00a0 A prime number is a positive integer such that it is only divisible by one and itself.\u00a0 In light of this definition, the number one is not considered a prime.<\/p>\n

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Though the properties of prime numbers are well defined, they are still very much mysterious.\u00a0\u00a0\u00a0 For instance, it becomes very ambiguous to calculate where a prime number will occur in a set of very large integers in a sequence.\u00a0 \u00a0Around 300 B.C., Euclid proved that there exists infinitely many prime numbers.\u00a0 This was not intuitive at first, because as numbers become larger, prime numbers become more \u201crare\u201d.\u00a0 His proof is simple.\u00a0 Basically, he assumed that there exists a last prime number \"P\". \u00a0\u00a0Then, he further reasoned, that \"P\" and all of the prime numbers less than \"P\" multiplied together form a composite number \"N\".\u00a0\u00a0 Therefore, he added one to \"N\", \"N+1\", to form a new integer that is not divisible by any prime number less than it.\u00a0 Consequently, this shows that \"N+1\" is a prime number by definition, which is proof by contradiction that there exist infinitely many primes.<\/p>\n

Many famous open problems in Number Theory (a major subfield of mathematics) are related to the nature of prime numbers. \u00a0\u00a0The nineteenth century French mathematician Marie-Sophie Germain was an originator of one such problem.\u00a0 \u00a0\u00a0She postulated a conjecture that was related to a special set of prime numbers that later became known as Sophie-Germain Primes.\u00a0 A Sophie-Germain Prime is a prime number \"P\" such that \"2P+1\" is also a prime.\u00a0 For example, \"2\" is a prime and \"2(2)+1 \u00a0is also a prime, making \"2\" a Sophie-Germain Prime.\u00a0 \u00a0\u00a0Like Euclid with the regular prime numbers, the conjecture poses the question: \u00a0Are there infinitely many Sophie-Germain Primes?<\/p>\n

A quick search on the internet will yield the current research and computational results related to the conjecture. \u00a0\u00a0\u00a0It would be interesting to see the emergence of a clever elementary proof like the one Euclid provided for the infinitude of regular primes.\u00a0 One could imagine the elegance of such a solution.\u00a0\u00a0How would you approach it?<\/p>\n

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Numbers pervade our lives in many different venues.\u00a0 Prime numbers, in particular, weave their way into the very fabric of our daily existence.\u00a0\u00a0 From surfing the internet to pseudo-random number generators, primes are found ever present behind a multitude of … Continue reading →<\/span><\/a><\/p>\n

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