List of divisors. By definition, the number n is prime only if it is not evenly divisible by 2 and any integers other than 1 and itself. The above formula removes unnecessary steps and saves time: for example, after checking if a number is divisible by 3, there is no need to check if it is divisible by 9.
- The floor(x) function rounds x to the nearest integer less than or equal to x.
Learn about modular arithmetic. The operation "x mod y" (mod is short for the Latin word "modulo", meaning "module") means "divide x by y and find the remainder". In other words, in modular arithmetic, upon reaching a certain value, which is called module, the numbers "turn" back to zero. For example, a clock measures time in modulus 12: it shows 10, 11, and 12 o'clock and then returns to 1.
- Many calculators have a mod key. The end of this section shows how to manually calculate this function for large numbers.
Learn about the pitfalls of Fermat's Little Theorem. All numbers for which the test conditions are not met are composite, but the remaining numbers are only probably are considered simple. If you want to avoid incorrect results, look for n in the list of "Carmichael numbers" (composite numbers that pass the given test) and "pseudo prime numbers Farm" (these numbers correspond to the test conditions only for some values a).
If convenient, use the Miller-Rabin test. Although this method rather cumbersome for manual calculations, it is often used in computer programs. It provides acceptable speed and gives less errors than Fermat's method. A composite number will not be taken as a prime number if calculations are made for more than ¼ values a. If you randomly choose various meanings a and for all of them the test will give a positive result, we can assume with a fairly high degree of confidence that n is a prime number.
For large numbers, use modular arithmetic. If you don't have a mod calculator handy, or if your calculator isn't designed to handle such large numbers, use the power properties and modular arithmetic to make calculations easier. Below is an example for 3 50 (\displaystyle 3^(50)) mod 50:
- Rewrite the expression in a more convenient form: mod 50. When calculating manually, further simplifications may be necessary.
- (3 25 ∗ 3 25) (\displaystyle (3^(25)*3^(25))) mod 50 = mod 50 mod 50) mod 50. Here we have taken into account the property of modular multiplication.
- 3 25 (\displaystyle 3^(25)) mod 50 = 43.
- (3 25 (\displaystyle (3^(25)) mod 50 ∗ 3 25 (\displaystyle *3^(25)) mod 50) mod 50 = (43 ∗ 43) (\displaystyle (43*43)) mod 50.
- = 1849 (\displaystyle =1849) mod 50.
- = 49 (\displaystyle=49).
- Translation
The properties of prime numbers were first studied by mathematicians Ancient Greece. Mathematicians Pythagorean school(500 - 300 BC) were primarily interested in the mystical and numerological properties of prime numbers. They were the first to come up with ideas about perfect and friendly numbers.
A perfect number has its own divisors equal to itself. For example, the proper divisors of the number 6 are: 1, 2 and 3. 1 + 2 + 3 = 6. The divisors of the number 28 are 1, 2, 4, 7 and 14. Moreover, 1 + 2 + 4 + 7 + 14 = 28.
Numbers are called friendly if the sum of proper divisors of one number is equal to another, and vice versa - for example, 220 and 284. We can say that a perfect number is friendly to itself.
By the time of the appearance of the work of Euclid's "Beginnings" in 300 BC. several have already been proven important facts about prime numbers. In Book IX of the Elements, Euclid proved that there are an infinite number of prime numbers. By the way, this is one of the first examples of the use of proof by contradiction. He also proves the Basic Theorem of Arithmetic - every integer can be represented in a unique way as a product of prime numbers.
He also showed that if the number 2 n -1 is prime, then the number 2 n-1 * (2 n -1) will be perfect. Another mathematician, Euler, in 1747 was able to show that all even perfect numbers can be written in this form. To this day, it is not known whether odd perfect numbers exist.
In the year 200 B.C. The Greek Eratosthenes came up with an algorithm for finding prime numbers called the Sieve of Eratosthenes.
And then there was a big break in the history of the study of prime numbers associated with the Middle Ages.
The following discoveries were made already at the beginning of the 17th century by the mathematician Fermat. He proved Albert Girard's conjecture that any prime number of the form 4n+1 can be written in a unique way as a sum of two squares, and also formulated a theorem that any number can be represented as a sum of four squares.
He developed new method factorization of large numbers, and demonstrated it on the number 2027651281 = 44021 × 46061. He also proved Fermat's Little Theorem: if p is a prime number, then for any integer a, a p = a modulo p will be true.
This statement proves half of what was known as the "Chinese hypothesis" and dates back 2000 years earlier: an integer n is prime if and only if 2n-2 is divisible by n. The second part of the hypothesis turned out to be false - for example, 2341 - 2 is divisible by 341, although the number 341 is composite: 341 = 31 × 11.
Fermat's Little Theorem was the basis for many other results in number theory and methods for testing whether numbers are prime, many of which are still in use today.
Fermat corresponded extensively with his contemporaries, especially with a monk named Marin Mersenne. In one of his letters, he conjectured that numbers of the form 2 n + 1 will always be prime if n is a power of two. He tested this for n = 1, 2, 4, 8, and 16, and was sure that when n is not a power of two, the number was not necessarily prime. These numbers are called Fermat numbers, and it wasn't until 100 years later that Euler showed that the next number, 232 + 1 = 4294967297, is divisible by 641 and therefore not prime.
Numbers of the form 2 n - 1 have also been the subject of research, since it is easy to show that if n is composite, then the number itself is also composite. These numbers are called Mersenne numbers because he actively studied them.
But not all numbers of the form 2 n - 1, where n is prime, are prime. For example, 2 11 - 1 = 2047 = 23 * 89. This was first discovered in 1536.
For many years, numbers of this kind gave mathematicians the largest known primes. That the number M 19 was proved by Cataldi in 1588, and for 200 years was the largest known prime number, until Euler proved that M 31 is also prime. This record held for another hundred years, and then Lucas showed that M 127 is prime (and this is already a number of 39 digits), and after that, research continued with the advent of computers.
In 1952, the primeness of the numbers M 521 , M 607 , M 1279 , M 2203 and M 2281 was proved.
By 2005, 42 Mersenne primes had been found. The largest of them, M 25964951 , consists of 7816230 digits.
Euler's work had a huge impact on number theory, including prime numbers. He extended Fermat's Little Theorem and introduced the φ-function. Factorized the 5th Fermat number 2 32 +1, found 60 pairs of friendly numbers, and formulated (but failed to prove) the quadratic law of reciprocity.
He was the first to introduce the methods of mathematical analysis and developed the analytic theory of numbers. He proved that not only the harmonic series ∑ (1/n), but also a series of the form
1/2 + 1/3 + 1/5 + 1/7 + 1/11 +…
Obtained by the sum of quantities inverse to prime numbers, also diverges. The sum of the n terms of the harmonic series grows approximately like log(n), while the second series diverges more slowly, like log[ log(n) ]. This means that, for example, the sum of the reciprocals of all the prime numbers found to date will give only 4, although the series still diverges.
At first glance, it seems that prime numbers are distributed among integers rather randomly. For example, among the 100 numbers immediately before 10000000, there are 9 primes, and among the 100 numbers immediately after this value, there are only 2. But on large segments, prime numbers are distributed fairly evenly. Legendre and Gauss dealt with their distribution. Gauss once told a friend that in any free 15 minutes he always counts the number of primes in the next 1000 numbers. By the end of his life, he had counted all the prime numbers up to 3 million. Legendre and Gauss equally calculated that for large n the density of primes is 1/log(n). Legendre estimated the number of primes between 1 and n as
π(n) = n/(log(n) - 1.08366)
And Gauss - as a logarithmic integral
π(n) = / 1/log(t) dt
With an integration interval from 2 to n.
The statement about the density of primes 1/log(n) is known as the Prime Numbers Theorem. They tried to prove it throughout the 19th century, and Chebyshev and Riemann made progress. They connected it with the Riemann Hypothesis, a hitherto unproven conjecture about the distribution of zeros of the Riemann zeta function. The density of primes was simultaneously proved by Hadamard and de la Vallée-Poussin in 1896.
In the theory of prime numbers, there are still many unresolved questions, some of which are many hundreds of years old:
- twin prime hypothesis - about an infinite number of pairs of prime numbers that differ from each other by 2
- Goldbach's conjecture: any even number, starting from 4, can be represented as the sum of two prime numbers
- Is there an infinite number of prime numbers of the form n 2 + 1 ?
- is it always possible to find a prime number between n 2 and (n + 1) 2 ? (the fact that there is always a prime number between n and 2n was proved by Chebyshev)
- Is there an infinite number of Fermat primes? are there any Fermat primes after the 4th?
- is there an arithmetic progression of consecutive primes for any given length? for example, for length 4: 251, 257, 263, 269. The maximum length found is 26 .
- Is there an infinite number of sets of three consecutive primes in an arithmetic progression?
- n 2 - n + 41 is a prime number for 0 ≤ n ≤ 40. Is there an infinite number of such prime numbers? The same question for the formula n 2 - 79 n + 1601. These numbers are prime for 0 ≤ n ≤ 79.
- Is there an infinite number of prime numbers of the form n# + 1? (n# is the result of multiplying all prime numbers less than n)
- Is there an infinite number of prime numbers of the form n# -1 ?
- Is there an infinite number of prime numbers of the form n! +1?
- Is there an infinite number of prime numbers of the form n! - one?
- if p is prime, does 2 p -1 always not include among the factors of squared primes
- Does the Fibonacci sequence contain an infinite number of primes?
The largest twin prime numbers are 2003663613 × 2 195000 ± 1. They consist of 58711 digits and were found in 2007.
The largest factorial prime number (of the form n! ± 1) is 147855! - 1. It consists of 142891 digits and was found in 2002.
The largest primorial prime number (a number of the form n# ± 1) is 1098133# + 1.
In this article, we will study prime and composite numbers. First, we give definitions of prime and composite numbers, and also give examples. After that, we prove that there are infinitely many prime numbers. Next, we write a table of prime numbers, and consider the methods for compiling a table of prime numbers, we will especially carefully dwell on the method called the sieve of Eratosthenes. In conclusion, we highlight the main points that need to be taken into account when proving that a given number is prime or composite.
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Prime and Composite Numbers - Definitions and Examples
The concepts of prime numbers and composite numbers refer to those that are greater than one. Such integers, depending on the number of their positive divisors, are divided into prime and composite numbers. So to understand definitions of prime and composite numbers, you need to have a good idea of \u200b\u200bwhat divisors and multiples are.
Definition.
prime numbers are integers, greater than one, that have only two positive divisors, namely themselves and 1 .
Definition.
Composite numbers are integers greater than one that have at least three positive divisors.
Separately, we note that the number 1 does not apply to either prime or composite numbers. The unit has only one positive divisor, which is the number 1 itself. This distinguishes the number 1 from all other positive integers that have at least two positive divisors.
Considering that positive integers are , and that the unit has only one positive divisor, other formulations of the sounded definitions of prime and composite numbers can be given.
Definition.
Prime numbers are natural numbers that have only two positive divisors.
Definition.
Composite numbers are natural numbers that have more than two positive divisors.
Note that every positive integer greater than one is either a prime number or a composite number. In other words, there is not a single integer that is neither prime nor composite. This follows from the divisibility property, which says that the numbers 1 and a are always divisors of any integer a.
Based on the information in the previous paragraph, we can give the following definition of composite numbers.
Definition.
Natural numbers that are not prime are called constituent.
Let's bring examples of prime and composite numbers.
As examples of composite numbers, we give 6 , 63 , 121 and 6697 . This statement also needs an explanation. The number 6, in addition to positive divisors 1 and 6, also has divisors 2 and 3, since 6 \u003d 2 3, therefore 6 is really a composite number. The positive divisors of 63 are the numbers 1 , 3 , 7 , 9 , 21 and 63 . The number 121 is equal to the product of 11 11 , so its positive divisors are 1 , 11 and 121 . And the number 6697 is composite, since its positive divisors, in addition to 1 and 6697, are also the numbers 37 and 181.
In conclusion of this paragraph, I would also like to draw attention to the fact that prime numbers and coprime numbers are far from the same thing.
Prime number table
Prime numbers, for convenience further use, are written in a table called the prime number table. Below is prime number table up to 1 000 .
A logical question arises: “Why did we fill out the table of prime numbers only up to 1,000, is it not possible to make a table of all existing prime numbers”?
Let's answer the first part of this question first. For most problems that involve prime numbers, primes up to a thousand will suffice. In other cases, most likely, you will have to resort to some special solution techniques. Although, of course, we can table prime numbers up to an arbitrarily large finite positive integer, whether it be 10,000 or 1,000,000,000 , in the next paragraph we will talk about methods for compiling tables of prime numbers, in particular, we will analyze the method called.
Now let's look at the possibility (or rather, the impossibility) of compiling a table of all existing prime numbers. We can't make a table of all the primes because there are infinitely many primes. The last statement is a theorem that we will prove after the following auxiliary theorem.
Theorem.
The smallest positive divisor of a natural number greater than 1 other than 1 is a prime number.
Proof.
Let be a - natural number, greater than one, and b is the smallest positive non-one divisor of a . Let us prove that b is a prime number by contradiction.
Suppose b is a composite number. Then there is a divisor of the number b (let's denote it b 1 ), which is different from both 1 and b . If we also take into account that the absolute value of the divisor does not exceed the absolute value of the dividend (we know this from the properties of divisibility), then the condition 1
Since the number a is divisible by b by condition, and we said that b is divisible by b 1 , then the concept of divisibility allows us to talk about the existence of such integers q and q 1 that a=b q and b=b 1 q 1 , whence a= b 1 ·(q 1 ·q) . From it follows that the product of two integers is an integer, then the equality a=b 1 ·(q 1 ·q) indicates that b 1 is a divisor of the number a . Taking into account the above inequalities 1
Now we can prove that there are infinitely many prime numbers.
Theorem.
There are infinitely many prime numbers.
Proof.
Let's assume it's not. That is, suppose that there are only n primes, and these primes are p 1 , p 2 , …, p n . Let us show that we can always find a prime number different from those indicated.
Consider a number p equal to p 1 ·p 2 ·…·p n +1 . It is clear that this number is different from each of the primes p 1 , p 2 , …, p n . If the number p is prime, then the theorem is proved. If this number is composite, then, by virtue of the previous theorem, there is a prime divisor of this number (let's denote it p n+1 ). Let's show that this divisor does not coincide with any of the numbers p 1 , p 2 , …, p n .
If this were not so, then by the properties of divisibility, the product p 1 ·p 2 ·…·p n would be divisible by p n+1 . But the number p is also divisible by p n+1, equal to the sum p 1 ·p 2 ·…·p n +1. This implies that the second term of this sum, which is equal to one, must be divisible by p n+1, and this is impossible.
Thus, it is proved that a new prime number can always be found, which is not contained among any number of prime numbers given in advance. Therefore, there are infinitely many prime numbers.
So, due to the fact that there are infinitely many prime numbers, when compiling tables of prime numbers, they always limit themselves from above to some number, usually 100, 1,000, 10,000, etc.
Sieve of Eratosthenes
Now we will discuss ways of compiling tables of prime numbers. Suppose we need to make a table of prime numbers up to 100 .
The most obvious method for solving this problem is to sequentially check positive integers, starting with 2 and ending with 100 , for the presence of a positive divisor that is greater than 1 and less than the number being checked (from the properties of divisibility, we know that the absolute value of the divisor does not exceed the absolute value of the dividend, different from zero). If such a divisor is not found, then the number being checked is prime, and it is entered in the table of prime numbers. If such a divisor is found, then the number being checked is composite, it is NOT entered into the table of prime numbers. After that, there is a transition to the next number, which is similarly checked for the presence of a divisor.
Let's describe the first few steps.
We start with the number 2. The number 2 has no positive divisors other than 1 and 2 . Therefore, it is prime, therefore, we enter it in the table of prime numbers. Here it should be said that 2 is the smallest prime number. Let's move on to number 3. Its possible positive divisor other than 1 and 3 is 2 . But 3 is not divisible by 2, therefore, 3 is a prime number, and it also needs to be entered in the table of prime numbers. Let's move on to number 4. Its positive divisors other than 1 and 4 can be 2 and 3 , let's check them. The number 4 is divisible by 2, therefore, 4 is a composite number and does not need to be entered in the table of prime numbers. Note that 4 is the smallest composite number. Let's move on to number 5. We check if at least one of the numbers 2 , 3 , 4 is its divisor. Since 5 is not divisible by either 2, or 3, or 4, it is prime, and it must be written in the table of prime numbers. Then there is a transition to the numbers 6, 7, and so on up to 100.
This approach to compiling a table of primes is far from ideal. One way or another, he has the right to exist. Note that with this method of constructing a table of integers, you can use divisibility criteria, which will slightly speed up the process of finding divisors.
There is a more convenient way to compile a table of primes called . The word “sieve” present in the name is not accidental, since the actions of this method help, as it were, to “sift” through the sieve of Eratosthenes integers, large units, in order to separate simple from compound ones.
Let's show the sieve of Eratosthenes in action when compiling a table of prime numbers up to 50.
First, we write down the numbers 2, 3, 4, ..., 50 in order.
The first number written 2 is prime. Now from the number 2 we sequentially move to the right by two numbers and cross out these numbers until we get to the end of the compiled table of numbers. So all numbers that are multiples of two will be crossed out.
The first non-crossed out number after 2 is 3 . This number is prime. Now, from the number 3, we sequentially move to the right by three numbers (taking into account the already crossed out numbers) and cross them out. So all numbers that are multiples of three will be crossed out.
The first non-crossed out number after 3 is 5 . This number is prime. Now, from the number 5, we sequentially move to the right by 5 numbers (we also take into account the numbers crossed out earlier) and cross them out. So all numbers that are multiples of five will be crossed out.
Next, we cross out numbers that are multiples of 7, then multiples of 11, and so on. The process ends when there are no numbers left to cross out. Below is a completed table of primes up to 50 obtained using the sieve of Eratosthenes. All uncrossed numbers are prime, and all crossed out numbers are composite.
Let's formulate and prove a theorem that will speed up the process of compiling a table of prime numbers using the sieve of Eratosthenes.
Theorem.
The least positive non-one divisor of a composite number a does not exceed , where is from a .
Proof.
Let b denote the smallest divisor of the composite number a that differs from unity (the number b is prime, which follows from the theorem proved at the very beginning of the previous paragraph). Then there is an integer q such that a=b q (here q is a positive integer, which follows from the rules of multiplication of integers), and (when b>q, the condition that b is the smallest divisor of a is violated, since q is also a divisor of a due to the equality a=q b ). Multiplying both sides of the inequality by a positive and greater than one integer b (we are allowed to do this), we obtain , whence and .
What does the proved theorem give us regarding the sieve of Eratosthenes?
First, the deletion of composite numbers that are multiples of a prime number b should begin with a number equal to (this follows from the inequality ). For example, crossing out numbers that are multiples of two should start with the number 4, multiples of three - with the number 9, multiples of five - with the number 25, and so on.
Secondly, the compilation of a table of prime numbers up to the number n using the sieve of Eratosthenes can be considered complete when all composite numbers that are multiples of prime numbers not exceeding are crossed out. In our example, n=50 (because we are tabulating primes up to 50 ) and , so the sieve of Eratosthenes must weed out all composite multiples of the primes 2 , 3 , 5 and 7 that do not exceed the arithmetic square root of 50 . That is, we no longer need to search and cross out numbers that are multiples of prime numbers 11 , 13 , 17 , 19 , 23 and so on up to 47 , since they will already be crossed out as multiples of smaller prime numbers 2 , 3 , 5 and 7 .
Is this number prime or composite?
Some tasks require finding out whether a given number is prime or composite. In the general case, this task is far from simple, especially for numbers whose record consists of a significant number of characters. In most cases, you have to look for some specific way to solve it. However, we will try to give direction to the train of thought for simple cases.
Undoubtedly, one can try to use divisibility criteria to prove that a given number is composite. If, for example, some criterion of divisibility shows that the given number is divisible by some positive integer greater than one, then the original number is composite.
Example.
Prove that the number 898 989 898 989 898 989 is composite.
Solution.
The sum of the digits of this number is 9 8+9 9=9 17 . Since the number equal to 9 17 is divisible by 9, then by the criterion of divisibility by 9 it can be argued that the original number is also divisible by 9. Therefore, it is composite.
A significant drawback of this approach is that the criteria for divisibility do not allow us to prove the simplicity of a number. Therefore, when checking a number for whether it is prime or composite, you need to proceed differently.
The most logical approach is to enumerate all possible divisors of a given number. If none of the possible divisors is a true divisor of a given number, then that number is prime; otherwise, it is composite. From the theorems proved in the previous paragraph, it follows that the divisors of a given number a must be sought among prime numbers not exceeding . Thus, the given number a can be successively divided by prime numbers (which are convenient to take from the table of prime numbers), trying to find the divisor of the number a. If a divisor is found, then the number a is composite. If among the prime numbers not exceeding , there is no divisor of the number a, then the number a is prime.
Example.
Number 11 723 simple or compound?
Solution.
Let's find out to what prime number the divisors of the number 11 723 can be. For this, we estimate .
It is quite obvious that 
, since 200 2 \u003d 40 000, and 11 723<40 000
(при необходимости смотрите статью number comparison). Thus, the possible prime divisors of 11,723 are less than 200. This already greatly simplifies our task. If we did not know this, then we would have to sort through all the prime numbers not up to 200, but up to the number 11 723 .
If desired, you can estimate more accurately. Since 108 2 \u003d 11 664, and 109 2 \u003d 11 881, then 108 2<11 723<109 2
, следовательно, 
. Thus, any of the primes less than 109 is potentially a prime divisor of the given number 11,723.
Now we will sequentially divide the number 11 723 into prime numbers 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 , 101 , 103 , 107 . If the number 11 723 is divided entirely by one of the written prime numbers, then it will be composite. If it is not divisible by any of the written prime numbers, then the original number is prime.
We will not describe this whole monotonous and monotonous process of division. Let's just say that 11 723
Prime numbers are one of the most interesting mathematical phenomena that has attracted the attention of scientists and ordinary citizens for more than two millennia. Despite the fact that we now live in the age of computers and the most modern information programs, many mysteries of prime numbers have not yet been solved, there are even those that scientists do not know how to approach.
Prime numbers are, as is known from the course of elementary arithmetic, those that are divisible without a remainder only by one and itself. By the way, if a natural number is divisible, in addition to those listed above, by another number, then it is called composite. One of the most famous theorems says that any composite number can be represented as the only possible product of prime numbers.
A few interesting facts. First, the unit is unique in the sense that, in fact, it does not belong to either prime or composite numbers. At the same time, in the scientific community it is still customary to attribute it to the first group, since formally it fully satisfies its requirements.
Secondly, the only even number that has crept into the “prime numbers” group is, of course, two. Any other even number simply cannot get here, since by definition, besides itself and one, it is also divisible by two.
Prime numbers, the list of which, as mentioned above, can begin with one, are an infinite series, as infinite as the series of natural numbers. Based on the fundamental theorem of arithmetic, one can come to the conclusion that prime numbers are never interrupted and never end, since otherwise the series of natural numbers would inevitably be interrupted.
Prime numbers do not appear randomly in the natural series, as it might seem at first glance. After carefully analyzing them, you can immediately notice several features, the most curious of which are associated with the so-called "twin" numbers. They are called so because, in some incomprehensible way, they ended up next to each other, separated only by an even delimiter (five and seven, seventeen and nineteen).
If you look closely at them, you will notice that the sum of these numbers is always a multiple of three. Moreover, when dividing by a triple of the left fellow, the remainder always remains a two, and the right one - one. In addition, the very distribution of these numbers along the natural series can be predicted if this entire series is represented in the form of oscillatory sinusoids, the main points of which are formed when the numbers are divided by three and two.
Prime numbers are not only an object of close scrutiny by mathematicians around the world, but have long been successfully used in compiling various series of numbers, which is the basis, including for ciphergraphy. At the same time, it should be recognized that a huge number of mysteries associated with these wonderful elements are still waiting to be solved, many questions have not only philosophical, but also practical significance.
A prime number is a natural number that is only divisible by itself and one.
The rest of the numbers are called composite.
Simple natural numbers
But not all natural numbers are prime.
Simple natural numbers are only those that are divisible only by themselves and by one.
Examples of prime numbers:
2; 3; 5; 7; 11; 13;...
Simple integers
It follows that only natural numbers are prime numbers.
This means that prime numbers are necessarily natural.
But all natural numbers are also integers.
Thus, all prime numbers are integers.
Examples of prime numbers:
2; 3; 5; 7; 11; 13; 17; 19; 23;...
Even prime numbers
There is only one even prime number, and that is two.
All other prime numbers are odd.
Why can't an even number greater than two be a prime number?
But because any even number greater than two will be divisible by itself, not by one, but by two, that is, such a number will always have three divisors, and possibly more.
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