Math textbooks are sometimes difficult to grasp. The dry and clear language of the authors is not always available for understanding. And the topics there are always interconnected, mutually emerging. To master one topic, you have to raise a number of previous ones, and sometimes turn over the entire textbook. Difficult? Yes. And let's take a chance to circumvent these difficulties and try to find a not entirely standard approach to the topic. Let's make a sort of excursion into the country of numbers. The definition, however, we still leave the same, because the rules of mathematics can not be canceled. So, mutually prime numbers are natural numbers, with a common divisor equal to one. This is clear? Quite.
For a more illustrative example, let's take the numbers 6 and 13. Both of them are divisible by one (mutually simple). But the numbers 12 and 14 cannot be such, since they are divided not only by 1, but also by 2. The following numbers - 21 and 47 also do not fit into the category of "relatively prime numbers": they can be divided not only by 1, but also at 7.
Mutually prime numbers are denoted as follows: ( a , y) = 1.
It can even be said simpler: the common divisor (largest) here is equal to one.
Why do we need such knowledge? There are enough reasons.
Mutually prime numbers are included in some encryption systems. Those who work with Hill ciphers or with Caesar's permutation system understand: without this knowledge, nowhere. If you have heard about pseudo-random number generators, then you are unlikely to deny it: mutually prime numbers are also used there.
Now let's talk about ways to get these numbers. Simple numbers , as you know, can have only two divisors: they are divisible by themselves and by one. Say, 11, 7, 5, 3 - the numbers are prime, but 9 is not, because this number is already divisible by 9, 3, and 1.
And if a is a prime and y from the set {1, 2, ... a - 1}, then guaranteedly ( a , y ) = 1, or mutually prime numbers - a and y .
Rather, this is not even an explanation, but a repetition or summing up of what has just been said.
Obtaining prime numbers is possible by Eratosthenes sieve, but for impressive numbers (billions, for example) this method is too long, but, unlike super-formulas, which sometimes make mistakes, are more reliable.
You can work by selecting y > a . For this, y is chosen so that the number is not divisible by a. To do this, a prime number is multiplied by a natural number and a quantity (say, p ) is added (or, on the contrary, subtracted), which is less than a :
y = p a + k
If, for example, a = 71, p = 3, q = 10, then, accordingly, here will be equal to 713. Another selection is possible, with degrees.
Compound numbers, unlike coprime ones, are divided into themselves, and by 1, and by other numbers (also without a remainder).
In other words, natural numbers (except unity) are divided into compound and simple.
Prime numbers are natural numbers that do not have non-trivial (different from the number and the unit itself) divisors. Especially important is their role in today's, modern, rapidly developing cryptography, due to which number theory, which was previously considered an extremely abstract discipline, has become so in demand: data protection algorithms are constantly being improved.
The largest prime number was found by ophthalmologist Martin Novak, who participated in the GIMPS project (distribution calculations) along with other enthusiasts, who totaled about 15 thousand. It took six long years to calculate. Two and a half dozen computers were deployed in the Novak eye clinic. The result of titanic work and perseverance was the number 225964951-1, with a record in 7816230 decimal places. By the way, the record for the largest number was set six months before this discovery. And there were half a million less signs.
A genius who wants to name a number where the duration of the decimal record “jumps over” the ten millionth mark has a chance to get not only worldwide fame, but also $ 100,000. By the way, for the number that crossed the millionth mark, Nayan Khayratval received a smaller amount ($ 50,000).