The Prime numbers, Also called absolute primes, they are those natural numbers that are only divisible between themselves and 1. This category includes numbers such as: 2, 3, 5, 7, 11, 13, 17, 19, 23 and many more.
Instead, a composite number is divisible by itself, by 1, and at least one other number. We have, for example, 12, which is divisible by 1, 2, 4, 6 and 12. By convention, 1 is not included in the list of prime numbers or in the list of compounds..
Knowledge of prime numbers dates back to ancient times; the ancient Egyptians already used them and they were surely known long before.
These numbers are very important, since any natural number can be represented by the product of prime numbers, this representation being unique, except in the order of the factors.
This fact is fully established in a theorem called The fundamental theorem of arithmetic, which states that the numbers that are not prime, are necessarily made up of products of numbers that are.
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Here are the main characteristics of prime numbers:
-They are infinite, since no matter how large a prime number is, you can always find a greater one.
-If a prime number p does not divide exactly to another number to, it is then said that p Y to they are cousins to each other. When this happens, the only common divisor that both have is 1.
It is not necessary to to be absolute cousin. For example, 5 is prime, and although 12 is not, both numbers are prime to each other, since both have 1 as a common divisor.
-When a prime number p divide to a power of number n, also divide n. Let's consider 100, which is a power of 10, specifically 10two. It happens that 2 divides both 100 and 10.
-All prime numbers are odd except for 2, therefore its last digit is 1, 3, 7 or 9. 5 is not included, because although it is odd and prime, it is never the final digit of another prime number. In fact all numbers that end in 5 are multiples of this and therefore are not prime.
-Yes p is prime and divisor of the product of two numbers a.b, then p divide one of them. For example, the prime number 3 divides the product 9 x 11 = 99, since 3 is a divisor of 9.
The primality is the name given to the quality of being prime. Well, the French mathematician Pierre de Fermat (1601-1665) found a way to verify the primality of a number, in the so-called Fermat's little theorem, That says so:
"Given a prime natural number p and any natural number to greater than 0, it is true that top - to is a multiple of p, as long as p be cousin ".
We can corroborate this using small numbers, for example suppose that p = 4, that we already know that it is not prime and a = 6:
64 - 6 = 1296 - 6 = 1290
The number 1290 is not divisible exactly by 4, therefore 4 is not a prime number.
Let's do the test now with p = 5, which is prime and a = 6:
65 - 6 = 7766 - 6 = 7760
7760 is divisible by 5, since any number that ends in 0 or 5 is. In fact 7760/5 = 1554. Since Fermat's little theorem holds, we can ensure that 5 is a prime number.
The proof through the theorem is effective and direct with small numbers, in which the operation is easy to perform, but what to do if we are asked to find out the primality of a large number?
In that case, the number is successively divided among all the smaller prime numbers, until an exact division is found or the quotient is less than the divisor.
If any division is exact, it means that the number is composite and if the quotient is less than the divisor, it means that the number is prime. We will put it into practice in solved exercise 2.
There are infinite prime numbers and there is no single formula to determine them. However, looking at some prime numbers like these:
3, 7, 31, 127 ...
It is observed that they are of the form 2n - 1, with n = 2, 3, 5, 7, 9 ... We make sure of it:
twotwo - 1 = 4 - 1 = 3; two3 - 1 = 8 - 1 = 7; two5 - 1 = 32 - 1 = 31; two7 - 1 = 128 - 1 = 127
But we cannot assure you that in general 2n - 1 is prime, because there are some values of n for which it does not work, for example 4:
two4 - 1 = 16 - 1 = 15
And the number 15 is not prime, since it ends in 5. However, one of the largest known prime numbers, found by computer calculations, is of the form 2n - 1 with:
n = 57,885,161
The Mersenne formula assures us that 2p - 1 is always prime, as long as p be cousin too. For example, 31 is prime, so 2 is safe31 - 1 is also:
two31 - 1 = 2,147,483,647
However, the formula allows you to determine only some prime numbers, not all.
The following polynomial allows finding prime numbers as long as n is between 0 and 39:
P (n) = ntwo + n + 41
Later, in the solved exercises section there is an example of its use.
Eratosthenes was an Ancient Greek physicist and mathematician who lived in the 3rd century BC. He devised a graphical method of finding the prime numbers that we can put into practice with small numbers, it is called the Eratosthenes sieve (a sieve is like a sieve).
-The numbers are placed in a table like the one shown in the animation.
-Then the even numbers are crossed out, except for 2, which we know is prime. All others are multiples of this and therefore are not prime.
-The multiples of 3, 5, 7 and 11 are also marked, excluding all of them because we know that they are prime.
-The multiples of 4, 6, 8, 9 and 10 are already marked, because they are compounds and therefore multiples of some of the indicated primes.
-Lastly, the numbers left unmarked are prime.
Using the Euler polynomial for prime numbers, find 3 numbers greater than 100.
This is the polynomial that Euler proposed to find prime numbers, which works for values of n between 0 and 39.
P (n) = ntwo + n + 41
By trial and error we select a value of n, for example n = 8:
P (8) = 8two + 8 + 41 = 113
Since n = 8 produces a prime number greater than 100, then we evaluate the polynomial for n = 9 and n = 10:
P (9) = 9two + 9 + 41 = 131
P (10) = 10two + 10 + 41 = 151
Find out if the following numbers are prime:
a) 13
b) 191
The 13 is small enough to use Fermat's little theorem and the help of the calculator.
We use a = 2 so that the numbers are not too large, although a = 3, 4 or 5 can also be used:
two13 - 2 = 8190
8190 is divisible by 2, since it is even, therefore 13 is prime. The reader can corroborate it by doing the same test with a = 3.
191 is too big to prove with the theorem and a common calculator, but we can find out the division between each prime number. We omit dividing by 2 because 191 is not even and the division will not be exact or the quotient less than 2.
We try to divide by 3:
191/3 = 63,666 ...
And it does not give exact, nor is the quotient less than the divisor (63,666… is greater than 3)
We continue thus trying to divide 191 between the primes 5, 7, 11, 13 and the exact division is not reached, nor the quotient less than the divisor. Until it is divided by 17:
191/17 = 11, 2352 ...
Since it is not exact and 11,2352… is less than 17, the number 191 is a prime.
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