The article deals with the concepts of prime and composite numbers. Definitions of such numbers with examples are given. We give a proof that the number of primes is unlimited and make an entry in the table of primes using the method of Eratosthenes. Proofs will be given as to whether a number is prime or composite.
Yandex.RTB R-A-339285-1
Prime and Composite Numbers - Definitions and Examples
Prime and composite numbers are classified as positive integers. They must be greater than one. Divisors are also divided into simple and compound. To understand the concept of composite numbers, it is necessary to first study the concepts of divisors and multiples.
Definition 1
Prime numbers are integers that are greater than one and have two positive divisors, that is, themselves and 1.
Definition 2
Composite numbers are integers that are greater than one and have at least three positive divisors.
One is neither a prime nor a composite number. It has only one positive divisor, so it is different from all other positive numbers. All positive integers are called natural, that is, used in counting.
Definition 3
prime numbers are natural numbers that have only two positive divisors.
Definition 4
Composite number- this natural number that has more than two positive divisors.
Any number greater than 1 is either prime or composite. From the property of divisibility, we have that 1 and the number a will always be divisors for any number a, that is, it will be divisible by itself and by 1. We give the definition of integers.
Definition 5
Natural numbers that are not prime are called composite numbers.
Prime numbers: 2, 3, 11, 17, 131, 523. They are divisible only by themselves and by 1. Composite numbers: 6, 63, 121, 6697. That is, the number 6 can be decomposed into 2 and 3, and 63 into 1, 3, 7, 9, 21, 63, and 121 into 11, 11, that is, its divisors will be 1, 11, 121. The number 6697 will decompose into 37 and 181. Note that the concepts of prime numbers and relatively prime numbers are different concepts.
To make it easier to use prime numbers, you need to use a table:
A table for all existing natural numbers is unrealistic, since there are an infinite number of them. When the numbers reach sizes of 10000 or 1000000000, then you should think about using the sieve of Eratosthenes.
Consider a theorem that explains the last statement.
Theorem 1
The smallest positive divisor of a natural number greater than 1 other than 1 is a prime number.
Proof 1
Assume that a is a natural number greater than 1, b is the smallest non-one divisor of a. We must prove that b is a prime number using the contradiction method.
Let's say b is a composite number. From here we have that there is a divisor for b , which is different from 1 as well as from b . Such a divisor is denoted as b 1 . It is necessary that condition 1< b 1 < b has been completed.
It can be seen from the condition that a is divisible by b, b is divisible by b 1, which means that the concept of divisibility is expressed in this way: a = b q and b = b 1 q 1 , whence a = b 1 (q 1 q) , where q and q 1 are integers. According to the rule of multiplication of integers, we have that the product of integers is an integer with an equality of the form a = b 1 · (q 1 · q) . It can be seen that b 1 is the divisor of a. Inequality 1< b 1 < b not matches, because we get that b is the smallest positive non-1 divisor of a.
Theorem 2
There are infinitely many prime numbers.
Proof 2
Suppose we take a finite number of natural numbers n and denote as p 1 , p 2 , … , p n . Let's consider a variant of finding a prime number different from the indicated ones.
Consider the number p, which is equal to p 1 , p 2 , … , p n + 1 . It does not equal each of the numbers corresponding to primes of the form p 1 , p 2 , … , p n . The number p is prime. Then the theorem is considered to be proved. If it is composite, then we need to take the notation p n + 1 and show divisor mismatch with any of p 1 , p 2 , … , p n .
If this were not so, then, based on the divisibility property of the product p 1 , p 2 , … , p n , we get that it would be divisible by p n + 1 . Note that the expression p n + 1 the number p is divided equals the sum p 1 , p 2 , … , p n + 1 . We get that the expression p n + 1 the second term of this sum, which is equal to 1, must be divided, but this is impossible.
It can be seen that any prime number can be found among any number of given primes. It follows that there are infinitely many prime numbers.
Since there are a lot of prime numbers, the tables are limited to numbers 100, 1000, 10000 and so on.
When compiling a table of prime numbers, one should take into account the fact that such a task requires a sequential check of numbers, starting from 2 to 100. If there is no divisor, it is recorded in the table; if it is composite, then it is not entered in the table.
Let's consider step by step.
If you start with the number 2, then it has only 2 divisors: 2 and 1, which means that it can be entered in the table. Also with the number 3 . The number 4 is composite, it should be decomposed into 2 and 2. The number 5 is prime, which means it can be fixed in the table. Do this up to the number 100.
This method uncomfortable and long. You can make a table, but you have to spend a large number of time. It is necessary to use divisibility criteria, which will speed up the process of finding divisors.
The method using the sieve of Eratosthenes is considered the most convenient. Let's take a look at the tables below. To begin with, the numbers 2, 3, 4, ..., 50 are written.
Now you need to cross out all the numbers that are multiples of 2. Make sequential strikethrough. We get a table of the form:
Let's move on to crossing out numbers that are multiples of 5. We get:
We cross out the numbers that are multiples of 7, 11. Finally the table looks like
Let us pass to the formulation of the theorem.
Theorem 3
The smallest positive and non-1 divisor of the base number a does not exceed a , where a is the arithmetic root of the given number.
Proof 3
It is necessary to designate b smallest divisor composite number a. There is an integer q , where a = b · q , and we have that b ≤ q . An inequality of the form b > q because the condition is violated. Both sides of the inequality b ≤ q should be multiplied by any positive number b not equal to 1 . We get that b b ≤ b q , where b 2 ≤ a and b ≤ a .
It can be seen from the proved theorem that the deletion of numbers in the table leads to the fact that it is necessary to start with a number that is equal to b 2 and satisfies the inequality b 2 ≤ a . That is, if you cross out numbers that are multiples of 2, then the process starts from 4, and those that are multiples of 3 start from 9, and so on up to 100.
Compiling such a table using Eratosthenes' theorem says that when all composite numbers are crossed out, there will remain prime ones that do not exceed n. In the example where n = 50 , we have that n = 50 . From here we get that the sieve of Eratosthenes sifts out all composite numbers that are not greater in value than the value of the root of 50. The search for numbers is done by crossing out.
Before solving, it is necessary to find out whether the number is prime or composite. Divisibility criteria are often used. Let's look at this in the example below.
Example 1
Prove that 898989898989898989 is a composite number.
Solution
The sum of the digits of the given number is 9 8 + 9 9 = 9 17 . So the number 9 17 is divisible by 9, based on the sign of divisibility by 9. It follows that it is composite.
Such signs are not able to prove the primeness of a number. If verification is needed, other steps should be taken. Most suitable way- It's a slew of numbers. During the process, prime and composite numbers can be found. That is, numbers in value should not exceed a . That is, the number a must be decomposed into prime factors. if this is true, then the number a can be considered prime.
Example 2
Determine the composite or prime number 11723.
Solution
Now you need to find all divisors for the number 11723. Need to evaluate 11723 .
From here we see that 11723< 200 , то 200 2 = 40 000 , and 11 723< 40 000 . Получаем, что делители для 11 723 less than number 200 .
For a more accurate estimate of the number 11723, it is necessary to write the expression 108 2 = 11 664, and 109 2 = 11 881 , then 108 2 < 11 723 < 109 2 . It follows from this that 11723< 109 . Видно, что любое число, которое меньше 109 считается делителем для заданного числа.
When decomposing, we get that 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 are all prime numbers. This whole process can be depicted as a division by a column. That is, divide 11723 by 19. The number 19 is one of its factors, since we get division without a remainder. Let's depict the division by a column:
It follows that 11723 is a composite number, because in addition to itself and 1 it has a divisor 19 .
Answer: 11723 is a composite number.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
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.
Page navigation.
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.
Given that positive integers are , and that the unit has only one positive divisor, other formulations of the voiced 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 the convenience of their further use, are recorded in a table, which is called the table of prime numbers. 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 make a table of prime numbers up to an arbitrarily large final integer positive number, be it 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 is a natural number greater than one, and b is the least 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
Table of prime numbers from 1 to 10000. Table of prime numbers from 1 to 1000
Below is a table of prime numbers from 2 to 10000 (1229 pieces). Unit not included, sorry. Some feel that the unit is not included because… she can't be there. " A prime number is a number that has two divisors: one and the number itself."And the number 1 has only one divisor, it does not apply to either prime or composite numbers. (An explanatory note from Olga on 09/21/12) However, we remember that prime numbers are sometimes introduced like this: " A prime number is a number that is evenly divisible by one and itself."In this case, one is obviously a prime number.
A table of prime numbers from 2 to 1000. The table of prime numbers from 2 to 1000 is grayed out.
| 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 | 109 | 113 | 127 | 131 | 137 | 139 | 149 | 151 |
| 157 | 163 | 167 | 173 | 179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 |
| 227 | 229 | 233 | 239 | 241 | 251 | 257 | 263 | 269 | 271 | 277 | 281 |
| 283 | 293 | 307 | 311 | 313 | 317 | 331 | 337 | 347 | 349 | 353 | 359 |
| 367 | 373 | 379 | 383 | 389 | 397 | 401 | 409 | 419 | 421 | 431 | 433 |
| 439 | 443 | 449 | 457 | 461 | 463 | 467 | 479 | 487 | 491 | 499 | 503 |
| 509 | 521 | 523 | 541 | 547 | 557 | 563 | 569 | 571 | 577 | 587 | 593 |
| 599 | 601 | 607 | 613 | 617 | 619 | 631 | 641 | 643 | 647 | 653 | 659 |
| 661 | 673 | 677 | 683 | 691 | 701 | 709 | 719 | 727 | 733 | 739 | 743 |
| 751 | 757 | 761 | 769 | 773 | 787 | 797 | 809 | 811 | 821 | 823 | 827 |
| 829 | 839 | 853 | 857 | 859 | 863 | 877 | 881 | 883 | 887 | 907 | 911 |
| 919 | 929 | 937 | 941 | 947 | 953 | 967 | 971 | 977 | 983 | 991 | 997 |
| 1009 | 1013 | 1019 | 1021 | 1031 | 1033 | 1039 | 1049 | 1051 | 1061 | 1063 | 1069 |
| 1087 | 1091 | 1093 | 1097 | 1103 | 1109 | 1117 | 1123 | 1129 | 1151 | 1153 | 1163 |
| 1171 | 1181 | 1187 | 1193 | 1201 | 1213 | 1217 | 1223 | 1229 | 1231 | 1237 | 1249 |
| 1259 | 1277 | 1279 | 1283 | 1289 | 1291 | 1297 | 1301 | 1303 | 1307 | 1319 | 1321 |
| 1327 | 1361 | 1367 | 1373 | 1381 | 1399 | 1409 | 1423 | 1427 | 1429 | 1433 | 1439 |
| 1447 | 1451 | 1453 | 1459 | 1471 | 1481 | 1483 | 1487 | 1489 | 1493 | 1499 | 1511 |
| 1523 | 1531 | 1543 | 1549 | 1553 | 1559 | 1567 | 1571 | 1579 | 1583 | 1597 | 1601 |
| 1607 | 1609 | 1613 | 1619 | 1621 | 1627 | 1637 | 1657 | 1663 | 1667 | 1669 | 1693 |
| 1697 | 1699 | 1709 | 1721 | 1723 | 1733 | 1741 | 1747 | 1753 | 1759 | 1777 | 1783 |
| 1787 | 1789 | 1801 | 1811 | 1823 | 1831 | 1847 | 1861 | 1867 | 1871 | 1873 | 1877 |
| 1879 | 1889 | 1901 | 1907 | 1913 | 1931 | 1933 | 1949 | 1951 | 1973 | 1979 | 1987 |
| 1993 | 1997 | 1999 | 2003 | 2011 | 2017 | 2027 | 2029 | 2039 | 2053 | 2063 | 2069 |
| 2081 | 2083 | 2087 | 2089 | 2099 | 2111 | 2113 | 2129 | 2131 | 2137 | 2141 | 2143 |
| 2153 | 2161 | 2179 | 2203 | 2207 | 2213 | 2221 | 2237 | 2239 | 2243 | 2251 | 2267 |
| 2269 | 2273 | 2281 | 2287 | 2293 | 2297 | 2309 | 2311 | 2333 | 2339 | 2341 | 2347 |
| 2351 | 2357 | 2371 | 2377 | 2381 | 2383 | 2389 | 2393 | 2399 | 2411 | 2417 | 2423 |
| 2437 | 2441 | 2447 | 2459 | 2467 | 2473 | 2477 | 2503 | 2521 | 2531 | 2539 | 2543 |
| 2549 | 2551 | 2557 | 2579 | 2591 | 2593 | 2609 | 2617 | 2621 | 2633 | 2647 | 2657 |
| 2659 | 2663 | 2671 | 2677 | 2683 | 2687 | 2689 | 2693 | 2699 | 2707 | 2711 | 2713 |
| 2719 | 2729 | 2731 | 2741 | 2749 | 2753 | 2767 | 2777 | 2789 | 2791 | 2797 | 2801 |
| 2803 | 2819 | 2833 | 2837 | 2843 | 2851 | 2857 | 2861 | 2879 | 2887 | 2897 | 2903 |
| 2909 | 2917 | 2927 | 2939 | 2953 | 2957 | 2963 | 2969 | 2971 | 2999 | 3001 | 3011 |
| 3019 | 3023 | 3037 | 3041 | 3049 | 3061 | 3067 | 3079 | 3083 | 3089 | 3109 | 3119 |
| 3121 | 3137 | 3163 | 3167 | 3169 | 3181 | 3187 | 3191 | 3203 | 3209 | 3217 | 3221 |
| 3229 | 3251 | 3253 | 3257 | 3259 | 3271 | 3299 | 3301 | 3307 | 3313 | 3319 | 3323 |
| 3329 | 3331 | 3343 | 3347 | 3359 | 3361 | 3371 | 3373 | 3389 | 3391 | 3407 | 3413 |
| 3433 | 3449 | 3457 | 3461 | 3463 | 3467 | 3469 | 3491 | 3499 | 3511 | 3517 | 3527 |
| 3529 | 3533 | 3539 | 3541 | 3547 | 3557 | 3559 | 3571 | 3581 | 3583 | 3593 | 3607 |
| 3613 | 3617 | 3623 | 3631 | 3637 | 3643 | 3659 | 3671 | 3673 | 3677 | 3691 | 3697 |
| 3701 | 3709 | 3719 | 3727 | 3733 | 3739 | 3761 | 3767 | 3769 | 3779 | 3793 | 3797 |
| 3803 | 3821 | 3823 | 3833 | 3847 | 3851 | 3853 | 3863 | 3877 | 3881 | 3889 | 3907 |
| 3911 | 3917 | 3919 | 3923 | 3929 | 3931 | 3943 | 3947 | 3967 | 3989 | 4001 | 4003 |
| 4007 | 4013 | 4019 | 4021 | 4027 | 4049 | 4051 | 4057 | 4073 | 4079 | 4091 | 4093 |
| 4099 | 4111 | 4127 | 4129 | 4133 | 4139 | 4153 | 4157 | 4159 | 4177 | 4201 | 4211 |
| 4217 | 4219 | 4229 | 4231 | 4241 | 4243 | 4253 | 4259 | 4261 | 4271 | 4273 | 4283 |
| 4289 | 4297 | 4327 | 4337 | 4339 | 4349 | 4357 | 4363 | 4373 | 4391 | 4397 | 4409 |
| 4421 | 4423 | 4441 | 4447 | 4451 | 4457 | 4463 | 4481 | 4483 | 4493 | 4507 | 4513 |
| 4517 | 4519 | 4523 | 4547 | 4549 | 4561 | 4567 | 4583 | 4591 | 4597 | 4603 | 4621 |
| 4637 | 4639 | 4643 | 4649 | 4651 | 4657 | 4663 | 4673 | 4679 | 4691 | 4703 | 4721 |
| 4723 | 4729 | 4733 | 4751 | 4759 | 4783 | 4787 | 4789 | 4793 | 4799 | 4801 | 4813 |
| 4817 | 4831 | 4861 | 4871 | 4877 | 4889 | 4903 | 4909 | 4919 | 4931 | 4933 | 4937 |
| 4943 | 4951 | 4957 | 4967 | 4969 | 4973 | 4987 | 4993 | 4999 | 5003 | 5009 | 5011 |
| 5021 | 5023 | 5039 | 5051 | 5059 | 5077 | 5081 | 5087 | 5099 | 5101 | 5107 | 5113 |
| 5119 | 5147 | 5153 | 5167 | 5171 | 5179 | 5189 | 5197 | 5209 | 5227 | 5231 | 5233 |
| 5237 | 5261 | 5273 | 5279 | 5281 | 5297 | 5303 | 5309 | 5323 | 5333 | 5347 | 5351 |
| 5381 | 5387 | 5393 | 5399 | 5407 | 5413 | 5417 | 5419 | 5431 | 5437 | 5441 | 5443 |
| 5449 | 5471 | 5477 | 5479 | 5483 | 5501 | 5503 | 5507 | 5519 | 5521 | 5527 | 5531 |
| 5557 | 5563 | 5569 | 5573 | 5581 | 5591 | 5623 | 5639 | 5641 | 5647 | 5651 | 5653 |
| 5657 | 5659 | 5669 | 5683 | 5689 | 5693 | 5701 | 5711 | 5717 | 5737 | 5741 | 5743 |
| 5749 | 5779 | 5783 | 5791 | 5801 | 5807 | 5813 | 5821 | 5827 | 5839 | 5843 | 5849 |
| 5851 | 5857 | 5861 | 5867 | 5869 | 5879 | 5881 | 5897 | 5903 | 5923 | 5927 | 5939 |
| 5953 | 5981 | 5987 | 6007 | 6011 | 6029 | 6037 | 6043 | 6047 | 6053 | 6067 | 6073 |
| 6079 | 6089 | 6091 | 6101 | 6113 | 6121 | 6131 | 6133 | 6143 | 6151 | 6163 | 6173 |
| 6197 | 6199 | 6203 | 6211 | 6217 | 6221 | 6229 | 6247 | 6257 | 6263 | 6269 | 6271 |
| 6277 | 6287 | 6299 | 6301 | 6311 | 6317 | 6323 | 6329 | 6337 | 6343 | 6353 | 6359 |
| 6361 | 6367 | 6373 | 6379 | 6389 | 6397 | 6421 | 6427 | 6449 | 6451 | 6469 | 6473 |
| 6481 | 6491 | 6521 | 6529 | 6547 | 6551 | 6553 | 6563 | 6569 | 6571 | 6577 | 6581 |
| 6599 | 6607 | 6619 | 6637 | 6653 | 6659 | 6661 | 6673 | 6679 | 6689 | 6691 | 6701 |
| 6703 | 6709 | 6719 | 6733 | 6737 | 6761 | 6763 | 6779 | 6781 | 6791 | 6793 | 6803 |
| 6823 | 6827 | 6829 | 6833 | 6841 | 6857 | 6863 | 6869 | 6871 | 6883 | 6899 | 6907 |
| 6911 | 6917 | 6947 | 6949 | 6959 | 6961 | 6967 | 6971 | 6977 | 6983 | 6991 | 6997 |
| 7001 | 7013 | 7019 | 7027 | 7039 | 7043 | 7057 | 7069 | 7079 | 7103 | 7109 | 7121 |
| 7127 | 7129 | 7151 | 7159 | 7177 | 7187 | 7193 | 7207 | 7211 | 7213 | 7219 | 7229 |
| 7237 | 7243 | 7247 | 7253 | 7283 | 7297 | 7307 | 7309 | 7321 | 7331 | 7333 | 7349 |
| 7351 | 7369 | 7393 | 7411 | 7417 | 7433 | 7451 | 7457 | 7459 | 7477 | 7481 | 7487 |
| 7489 | 7499 | 7507 | 7517 | 7523 | 7529 | 7537 | 7541 | 7547 | 7549 | 7559 | 7561 |
| 7573 | 7577 | 7583 | 7589 | 7591 | 7603 | 7607 | 7621 | 7639 | 7643 | 7649 | 7669 |
| 7673 | 7681 | 7687 | 7691 | 7699 | 7703 | 7717 | 7723 | 7727 | 7741 | 7753 | 7757 |
| 7759 | 7789 | 7793 | 7817 | 7823 | 7829 | 7841 | 7853 | 7867 | 7873 | 7877 | 7879 |
| 7883 | 7901 | 7907 | 7919 | 7927 | 7933 | 7937 | 7949 | 7951 | 7963 | 7993 | 8009 |
| 8011 | 8017 | 8039 | 8053 | 8059 | 8069 | 8081 | 8087 | 8089 | 8093 | 8101 | 8111 |
| 8117 | 8123 | 8147 | 8161 | 8167 | 8171 | 8179 | 8191 | 8209 | 8219 | 8221 | 8231 |
| 8233 | 8237 | 8243 | 8263 | 8269 | 8273 | 8287 | 8291 | 8293 | 8297 | 8311 | 8317 |
| 8329 | 8353 | 8363 | 8369 | 8377 | 8387 | 8389 | 8419 | 8423 | 8429 | 8431 | 8443 |
| 8447 | 8461 | 8467 | 8501 | 8513 | 8521 | 8527 | 8537 | 8539 | 8543 | 8563 | 8573 |
| 8581 | 8597 | 8599 | 8609 | 8623 | 8627 | 8629 | 8641 | 8647 | 8663 | 8669 | 8677 |
| 8681 | 8689 | 8693 | 8699 | 8707 | 8713 | 8719 | 8731 | 8737 | 8741 | 8747 | 8753 |
| 8761 | 8779 | 8783 | 8803 | 8807 | 8819 | 8821 | 8831 | 8837 | 8839 | 8849 | 8861 |
| 8863 | 8867 | 8887 | 8893 | 8923 | 8929 | 8933 | 8941 | 8951 | 8963 | 8969 | 8971 |
| 8999 | 9001 | 9007 | 9011 | 9013 | 9029 | 9041 | 9043 | 9049 | 9059 | 9067 | 9091 |
| 9103 | 9109 | 9127 | 9133 | 9137 | 9151 | 9157 | 9161 | 9173 | 9181 | 9187 | 9199 |
| 9203 | 9209 | 9221 | 9227 | 9239 | 9241 | 9257 | 9277 | 9281 | 9283 | 9293 | 9311 |
| 9319 | 9323 | 9337 | 9341 | 9343 | 9349 | 9371 | 9377 | 9391 | 9397 | 9403 | 9413 |
| 9419 | 9421 | 9431 | 9433 | 9437 | 9439 | 9461 | 9463 | 9467 | 9473 | 9479 | 9491 |
| 9497 | 9511 | 9521 | 9533 | 9539 | 9547 | 9551 | 9587 | 9601 | 9613 | 9619 | 9623 |
| 9629 | 9631 | 9643 | 9649 | 9661 | 9677 | 9679 | 9689 | 9697 | 9719 | 9721 | 9733 |
| 9739 | 9743 | 9749 | 9767 | 9769 | 9781 | 9787 | 9791 | 9803 | 9811 | 9817 | 9829 |
| 9833 | 9839 | 9851 | 9857 | 9859 | 9871 | 9883 | 9887 | 9901 | 9907 | 9923 | 9929 |
| 9931 | 9941 | 9949 | 9967 | 9973 | end of plate 🙂 ! | ||||||
Article rating:
Loading...