Mutually prime numbers. Basics

Textbooks of mathematics are sometimes difficult to comprehend. The dry and clear language of the authors is not always available for understanding. And the topics there are always interconnected, mutually flowing. To master one topic, one has to raise a number of previous ones, and sometimes even leaf through the entire textbook. Complicated? Yes. And let's dare to bypass these difficulties and try to find the topic is not quite a standard approach. Let's make a digression into the country of numbers. Definition, however, we still leave the same, because the rules of mathematics can not be canceled. So, relatively prime numbers are natural numbers, with a common divisor equal to one. It's clear? Completely.

For a more illustrative example, let's take the numbers 6 and 13. Both are divisible by one (relatively simple). But the numbers 12 and 14 - they can not be those, because they are divided not only into 1, but also to 2. The following numbers - 21 and 47 also do not fit into the category "mutually prime numbers": they can be divided not only by 1, but Also on 7.

Denote mutually prime numbers as follows: ( a , y) = 1.

It can even be said more simply: the common divisor (the largest) here is equal to one.
Why do we need such knowledge? There are enough reasons.

Mutually simple numbers are included in some encryption systems. Those who work with the ciphers of Hill or with the system of permutations of Caesar, understand: without this knowledge - anywhere. If you have heard about generators of pseudo-random numbers, you are unlikely to dare to deny that relatively prime numbers are used there.

Now let's talk about ways to get such numbers. Numbers are simple, as you know, can have only two divisors: they are divisible by themselves and by one. Say 11, 7, 5, 3 are simple numbers, but 9 is not, because this number is already divisible by 9, and by 3, and by 1.

And if a is a prime number and y is from the set {1, 2, ... a - 1}, then it is guaranteed ( a , y ) = 1, or relatively prime numbers - a and y .

This is, rather, not even an explanation, but a repetition or summarizing of what has just been said.

Getting prime numbers is possible with Eratosthenes's lattice, but for impressive numbers (billions, for example) this method is too long, but, unlike super formulas, which are sometimes wrong, more reliable.

You can work by selecting y > a . To do this, y is chosen so that the number on a is not divisible. For this, the prime number is multiplied by a natural number and the quantity (say, p ) that is less than a is added (or, conversely, subtracted)

Y = p a + k

If, for example, a = 71, p = 3, q = 10, then, respectively, y will be equal to 713. Another selection, with degrees, is also possible.

Compound numbers, unlike mutually simple ones, are divided into themselves, and to 1, and to other numbers (also without remainder).

In other words, natural numbers (except one) are divided into compound and simple numbers .

Prime numbers are natural numbers that do not have nontrivial divisors (distinct from the number and one). Particularly important is their role in today's modern, rapidly developing cryptography, thanks to which the theory of numbers, previously considered a discipline of the most abstract, has become so in demand: data protection algorithms are constantly being improved.

The largest prime number was found by the ophthalmologist Dr. Martin Novak, who participated in the GIMPS project (distributive calculations), along with other enthusiasts, who numbered about 15,000. The calculations took six long years. It involved two and a half dozen computers located in the eye clinic of Nowak. The result of the titanic work and perseverance was the number 225964951-1, with the writing in 7816230-decimal places. By the way, the record of the largest number was put half a year before this opening. And the signs there were half a million less.

A genius who wants to name a number where the length of the decimal record "jumps" the 10 millionth mark is a chance to get not only world fame, but also 100 000 dollars. By the way, Nyan Hiratwal received a smaller amount ($ 50,000) for the number that overcame the one million mark line.