PRIME NUMBERS A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 × 3, 1 × 1 × 3, etc. are all valid factorizations of 3. The property of being prime (or not) is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and \sqrt{n}. Algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of April 2014, the largest known prime number has 17,425,170 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known useful formula that sets apart all of the prime numbers from composites. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n. Many questions around prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals. From wikipedia.
Not at all Captain. PRIME NUMBERS FROM 1 TO 100 In total there are 25. Please Register or Log in to view the hidden image!
It is a simple idea that I'm still developing it. It explains the order of primes. Maybe with no relevance. Saludos.
SOME CHARACTERISTICS OF PRIME NUMBERS - All prime numbers are odd, except 2. - From 11 all prime numbers end in 1, 3, 7, 9. - Any even number can be expressed like the sum of two primes. PRIME NUMBERS FROM 1 TO 1000 In total there are 168. Please Register or Log in to view the hidden image! Ciao.
Wait, I almost finish the idea. CLASSIFICATION OF NUMBERS - C, COMPLEX: R, REAL AND IMAGINARY - REAL: Q, RATIONAL AND IRRATIONAL - RATIONAL: Z, INTEGERS AND FRACTIONS - FRACTIONS: PROPER, IMPROPER AND MIXED FRACTIONS - INTEGERS: N, NATURAL, ZERO AND NEGATIVE INTEGERS - NATURAL: EVEN AND ODD NUMBERS - NATURAL: PRIMES AND COMPOSITE
KILO NUMBERS This is a philosophical and mathematical point of view about the order that appear primes. As we know, from 11, all prime numbers end in 1, 3, 7, 9. Kilo (K) numbers are those composite numbers that end in 1, 3, 7, 9. From 1 to 100 there are 16 K numbers: 21, 51, 81, 91, 33, 63, 93, 27, 57, 77, 87, 9, 39, 49, 69, 99. For example: 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 ... - Prime Numbers in red. - K numbers in blue. The set of prime numbers and K numbers make the primes-K set: n1, n3, n7, n9, which have an internal difference of 2, 4, 2 that is repeated indefinitely. Prime numbers, by themselves, have an irregular order. If there was a logical sequence someone would have suggested it long ago. Sibilia
So, you discard all the numbers that are divisible by two, and then you discard the numbers that are divisible by any other prime, then you are left with new primes. Is this thread going anywhere?
My theory is that there isn't a theory. Question: Out of the numbers 1, 3, 7 and 9, which digit ends prime numbers most often? Answer: First 10'000: 3, 7, 9, 1 First 50'000: 3, 7, 1, 9 First 100'000: 7, 9, 3, 1 http://korn19.ch/coding/primes/ending.php
You could substitute the numbers with letters. The principle would still work. a is a prime b is a prime b squared, cubed etc are composite c is a prime bc is composite c squared, cubed etc are composite d is a prime db and dc are composite d squared, cubed etc are composite d squared, cubed etc times c are composite c squared, cubed etc times d are composite d squared, cubed etc times c squared, cubed etc are composite same as above with b and c; b and d; and b c and d. The more letters you have the greater the number of combinations.
I swim 24 laps usually three times each week.* I need to tell myself things to keep count. I noticed that if 3 is the first prime the second, 5, is two greater, then we have the fourth prime, 11, which is 4 greater than the third prime, 7. Does this happen again? I.e. the Nth prime is the (N-1)th prime plus their difference? (Excluding the obvious infinite number of paired primes.) While swimming, I guessed not.** If yes, does it happen an infinite number of times? I.e. is there another Nth = (N-n)th +(N-n), where (N-n)th is an earlier prime in the ordered list of primes and n is not 2. I.e. is 11 = 7+4 the only case? * Brazil is very nice to old people - cost me less than $100 per year for use of the heated enclosed pool and than includes the four times per year the pool's doctor looking to see if I have athletes foot etc. ** I could not do the mental testing while swimming a lap and when the lap number increased by one I had to drop the question to keep count of laps. (tell myself some new math story.) 24 is good number - lots of things I can tell my self like lap 16 is 2/3 done, lap 18 is 3/4ths done etc. I'm thinking about trying to do 36 laps, as there are lots of factors of that to help me keep track of what lap I'm on.
What are called "Twin Primes" differ by 2 but all prime pairs differ by an even number. Here are the first twin prime: 3,5 .. 5,7 .. 11,13 .. 17,19 ... up to at least: 3,756,801,695,685 × 2^2666,669 + & - 1, which is the largest pair known. It was discovered in 2011.* That is a number well beyond human comprehension, even just 2^2666,669 is and it is nearly 4 trillion times smaller! This pair may be the last or as most believe there is no "last pair" of twin primes. OK how about "delta 4 twin primes" ? Like: 3,7 .. 7,11 .. 19,23 ... because the recent progress can not prove there are an infinite number of "delta 2 twin prime" I'm nearly sure there is also no way to know if here is a largest pair of delta 4 twin primes . Probably that is also true about "delta 6 twin primes" like: 5,11 .. 7,13 .. 11,17... Extension /generalization of this idea to "delta trillion twin primes" may not be true, but again it may be? It all gives me a better feeling for fact that "infinity" on the number line is quite far from zero. * I'm wondering how many bitcoins, in 2011, the amount of computer time used to find the largest know twin primes would have generated? - That just goes to show, mathematicians are quite detached from the "real world" - live in their own closed tautology.
