natural numbers 1, 2, 3, 4, 5, 6, 7, ,...,, n, there are some special issues that are only divisible by unity and by themselves. These numbers are called prime numbers and from which are known to have produced a strange fascination among mathematicians. There are infinite, Euclid made the first known demonstration of its infinity around 300 BC, but its distribution, apparently random, remains a mystery.
Somehow, could say that these numbers are "one piece" and all other natural numbers can be constructed from them by a process called factoring. The first prime numbers less than one hundred are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73 , 79, 83, 89 and 97. Each of them can only be written as: 2 = 2, 3 = 3 = 29 ,..., 29 ,..., 67 = 67, ..., etc.. While the rest of natural numbers need to be expressed in terms of primes: 4 = 2x2, 9 = 3x3 6 = 3x2, 8 = 2x2x2, ..., 30 = 2x3x5, etc.
an important expression is known theorem called prime numbers gives us the number of prime numbers that exist to a certain number. Approximately, for numbers large enough, the expression is: number of primes = (number) / natural logarithm (number). Applying the formula (number) = 1000 we get 145 cousins, when in fact there are 168. For 5000 we get a little more, the expression gives us 587 and there are actually 669, and as we test larger numbers we get closer, but the numbers converge very slowly, for 1000 86.3%, 87.7% for 5000 and 90% for 50000. Lagunas
with no prime numbers:
Between 1 and 100 there are 25 prime numbers, as we have seen, and list observed groups of digits, a species lagoons with no prime numbers from 24 to 28 and 90 to 96. Between 100 and 200 there are 23 cousins, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149.151, 157, 163, 167, 173, 179, 181, 191.193, 197, 199. And as the gaps are 182 to 190. We may ask whether there are larger gaps between primes. At first glance, it seems that we will not find any of these gaps clearly with a sufficient amount of numbers, but it is not. can find as many as you want and the length you want, for it will use the following expression (may be many more): n! +2, from 2 to n. Let's see some examples: for n = 3, 3! = 3x2x1 = 6, 6 +2 = 8 and 6 +3 = 9. We found the first lake formed by 8 and 9. Continue with n = 4: 4! = 4x3x2x1 = 24, 24 +2 = 26, 24 and 24 +3 = 27 +4 = 28. We found three consecutive composite numbers, but with this expression we can find whatever we want, for example 101 numbers in a row (at least): 102! +2, 102! +3, 102! +3, ..., 102! +101.102! +102.
How many parts are made of numbers?
return to the title of the post, you can see the numbers as compounds formed by pieces of prime numbers. Any compound number, for example, 6 is equal the product of two primes 2x3, we can consider it as consisting of two parts, Part 2 and Part 3. Instead of prime numbers, as 7, are formed by only one piece. In a musical analogy the prime could be considered as a single primary harmonic, and composite number as a composition of cousins \u200b\u200bharmonics spectrum or decay form factor.
analyzing the factorization of a number as a product of primes, we could imagine any number is composed of many parts as its component prime factors. It is noted as a curiosity that numbers around 100 would be formed, as average of 2.7 for a product primes, around 1000 by a product of 2.96 primes, of 10000 and a product of 3.16 numbers, by 3.3 of 100000, 1000000; the by 3.42 and 3.64 of 10000000 by. We note that the number of "pieces" necessary to form any number increase very slowly, and this increase also decreases. It is somewhat amazing that while a 3-digit number needs to be factored three cousins \u200b\u200b( is made of three parts ), one of 10 places only need four ( is made of four parts ). course, speaking of these pieces are as diverse as 3 to 2000003, both are numbers cousins.
In a bizarre (and imaginary) quantum world consists of integers, it would be easy to find prime numbers . All composite numbers would be as a superposition of harmonic fuzzy prime while prime numbers appear clear and stable with a single configuration easily distinguishable. Some of this must occur owes Daniel Tammet, an autistic young English with an amazing capacity for numbers. When you think of them sees shapes, colors and textures that allow you to distinguish an amazing way. When multiplying two numbers you see two shadows at once there is a third shade that matches with the answer to the question. When you think of a number known to recognize him as prime or composite. I was watching the story on his life, his powers as a mathematician and his prodigious memory. His skills are amazing. Within a week, managed to learn, from scratch, enough Icelandic (language classified as very difficult) to keep well an interview on television in Iceland.
Somebody might seem that the study of prime numbers is of no use, of course is wrong (eye RSA encryption algorithm allows reliable transactions.) Any mathematical knowledge, however absurd we deem is related with many seemingly unrelated fields. Any advance in knowledge about prime numbers, for example, could be crucial to solve a problem most incredible field that comes to mind, both mathematical and physical. The reality is related and as the understanding we see that the knowledge we have of it is too.
A novel investigation of primes:
on prime numbers I remember reading an interesting novel entitled " Uncle Petros and Goldbach's conjecture ." The plot runs through the vicissitudes of a mathematician obsessed with checking the Goldbach conjecture famous on prime numbers, one of the oldest unsolved problems in mathematics. His statement is as follows: Any number greater than 2 can be written as a sum of two primes. I confess I did catch me like that has happened to countless readers. It is very entertaining and recommended.
... My thanks to Descartes page of the Ministry of Education, which has given me the calculations factoring large numbers I needed.
... I recommend to visit this magnificent page about prime numbers (in English).
Our friend Tito Eliatron sends us two very interesting links from your blog to a discussion of the mathematician, Fields Medal, Terry Tao : Part of the talk , second half. Thanks Tito.
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