For example, 91 can be factored using the identity above,. With this definition, every integer can be factored into prime integers, but the factorisation is only unique if primes in the factorisation are allowed to be replaced by their opposites. The following factorisation of the prime 5 involving the imaginary number i shows that primes have to be defined quite differently when working with complex numbers:.
Although primes were probably known to the Egyptians, the first known study of them occurs in the Elements of the Greek mathematician Euclid about BC. Euclid proved that every number can be factored uniquely into primes, and also proved that there are infinitely many prime numbers.
In , Goldbach famously conjectured that every even number greater than 2 is the sum of two prime numbers. Prime pairs stand out in the list of primes up to Mathematics is full of unsolved problems.
Wikipedia is a good place to find links to the present state of these and other outstanding problems. Both the Goldback Conjecture and the twin prime conjecture are thought to be true. The largest known prime. There is no known algorithm for generating arbitrarily large prime numbers. Prime numbers of this type are called Mersenne primes , and the index of 2 in such a prime must be a prime number. Applications of primes in security codes. One of the most important problems in everyday life is the secure transmission of information.
Prime numbers are used in computing as a means of encoding information so that it can be kept secure. Although in theory, every whole number is a product of primes, in practice it is very hard to find the actual factorisation of a very large number, even using high speed modern computers.
Computer scientists exploit this fact to build codes that are very hard to break, using very large prime numbers. The primes 3, 7, 11 form an arithmetic sequence of three primes.
An arithmetic sequence is a sequence that increases at each step by a common difference , which in this case is 4. The primes 5, 11, 17, 23, 29 form an arithmetic sequence of five primes — in this case the common difference is 6.
In , Terence Tao from Australia and Ben Green from the UK proved that there are arithmetic sequences of primes of any given length. For this and other work, Tao in became the first Australian to be awarded a Fields Medal, which is considered to be equivalent to a Nobel Prize in Mathematics..
Find an arithmetic sequence of six or more primes. The record at the time of writing is a sequence of 25 primes found in , but see Wikipedia for more details. Sequences of successive composite numbers. In contrast to these difficult questions about primes, the following exercise easily shows that there are sequences of arbitrary length consisting only of successive composite numbers.
Appendix — Proving the fundamental theorem of arithmetic. Here is a proof of the prime factorisation theorem. As noted in the module, it is the uniqueness that is difficult to prove. Three initial lemmas are needed, each of which is important in its own right.
Subtracting before finding the HCF. When finding the HCF of two numbers, it is often useful to subtract them. For example, when asked to find the HCF of 30 and 26, it is natural to subtract the two numbers and find instead the HCF of 26 and 4. Euclid shows in his Elements how to continue this subtraction procedure, with the numbers getting smaller at each step, until one of the numbers is zero. Then the other number is the HCF. Hence the HCF of 26 and 30 is 2.
Similarly the third step is either two subtractions of 2, or a single division. Starting from the second-last step and working upwards, we can express 2 as a sum of integer multiples of the original numbers 30 and This process is formalised in the following lemma to prove that it can be carried out in every case. The lemma is only stated in the case where the HCF of the two numbers is 1, because that is all that is needed on the later proof, but the exercise following the lemma quickly extends it to the general case.
Then there exist integers x and y so that. Thus by the induction hypothesis we can choose integers x and y so that. Let a and b be two whole numbers, not both zero, with HCF d. The prime number 5 is a divisor of Our experiences of factoring numbers should convince us that whenever is factored as a product, the prime 5 will be a divisor of one of the factors. For example, 5 is a divisor of one the factors in each factoring below:. The formal statement of this result is Lemma 3 below, which will provide us with the key step in proving the uniqueness of prime factorisation.
Its proof requires Lemma 2. Let p be a prime divisor of ab , where a and b are non-zero whole numbers. Suppose that p is not a divisor of a. Hence p is a divisor of b , because p is a divisor of both ab and pby. Let p be whole number greater than 1. The number 24 has an interesting property: it can be divided into whole equal parts in a relatively large number of ways. This means that a day can be divided into two equal parts of 12 h each, daytime and nighttime.
In a factory that works non-stop in 8-h shifts, each day is divided into exactly three shifts. If the circle is divided into two, three, four, ten, twelve, or thirty equal parts, each part will contain a whole number of degrees; and there are additional ways of dividing a circle that we did not mention.
In ancient times, dividing a circle into equal-sized sectors with high precision was necessary for various artistic, astronomical, and engineering purposes. With a compass and protractor as the only available instruments, division of a circle into equal sectors had great practical value. A whole number that can be written as the product of two smaller numbers is called a composite number. A number that cannot be broken down in this way is called a prime number.
The numbers. In fact, these are the first 10 prime numbers you can check this yourself, if you wish! Looking at this short list of prime numbers can already reveal a few interesting observations.
First, except for the number 2, all prime numbers are odd, since an even number is divisible by 2, which makes it composite. So, the distance between any two prime numbers in a row called successive prime numbers is at least 2.
In our list, we find successive prime numbers whose difference is exactly 2 such as the pairs 3,5 and 17, There are also larger gaps between successive prime numbers, like the six-number gap between 23 and 29; each of the numbers 24, 25, 26, 27, and 28 is a composite number. Another interesting observation is that in each of the first and second groups of 10 numbers meaning between 1—10 and 11—20 there are four prime numbers, but in the third group of 10 21—30 there are only two.
What does this mean? Do prime numbers become rarer as the numbers grow? Can anyone promise us that we will be able to keep finding more and more prime numbers indefinitely? Do not continue reading! Write all the numbers up to and mark the prime numbers. Check how many pairs with a difference of two are there.
Check how many prime numbers there are in each group of Can you find any patterns? Or does the list of prime numbers up to seem random to you? Prime numbers have occupied human attention since ancient times and were even associated with the supernatural.
Even today, in modern times, there are people trying to provide prime numbers with mystical properties. The idea that signals based on prime numbers could serve as a basis for communication with extraterrestrial cultures continues to ignite the imagination of many people to this day. It is commonly assumed that serious interest in prime numbers started in the days of Pythagoras. Pythagoras was an ancient Greek mathematician.
His students, the Pythagoreans—partly scientists and partly mystics—lived in the sixth century BC. They did not leave written evidence and what we know about them comes from stories that were passed down orally. Three hundred years later, in the third century BC, Alexandria in modern Egypt was the cultural capital of the Greek world.
Euclid Figure 1 , who lived in Alexandria in the days of Ptolemy the first, may be known to you from Euclidean geometry, which is named after him.
Euclidean geometry has been taught in schools for more than 2, years. But Euclid was also interested in numbers. This is a good place to say a few words about the concepts of theorem and mathematical proof. A theorem is a statement that is expressed in a mathematical language and can be said with certainty to be either valid or invalid.
To be more precise, this theorem claims that if we write a finite list of prime numbers, we will always be able to find another prime number that is not on the list. To prove this theorem, it is not enough to point out an additional prime number for a specific given list. For instance, if we point out 31 as a prime number outside the list of first 10 primes mentioned before, we will indeed show that that list did not include all prime numbers.
But perhaps by adding 31 we have now found all of the prime numbers, and there are no more? What we need to do, and what Euclid did 2, years ago, is to present a convincing argument why, for any finite list, as long as it may be, we can find a prime number that is not included in it. If you pick a number that is not composite, then that number is prime itself.
Otherwise, you can write the number you chose as a product of two smaller numbers. Prime numbers pop up in surprising places, check out why they are more than just math. To get you to come around to liking prime numbers we are going to stick to some very basic facts about them. And the first fact is simply: prime numbers are cool. In that book, aliens choose to send a long string of prime numbers as proof that their message is intelligent and not natural in origin, since primes are one thing that cannot change due to differences of psychology, lifestyle, or evolutionary history.
No matter what an advanced alien life-form looks or thinks like, if it understands the world around it, it almost certainly has the concept of a prime. But to truly understand the importance of prime numbers, we will have to go deeper. Most people are probably familiar with at least the basic idea of prime numbers. For those that need a refresher, however, here it is. Primes are the set of all numbers that can only be equally divided by 1 and themselves, with no other even division possible.
For example, numbers like 2, 3, 5, 7, and 11 are all prime numbers. If you looking for primes then, half of all possible numbers can be taken off the table right away the evens , along with all multiples of three, four, five, and so on.
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