The article discusses the concepts of prime and composite numbers. Definitions of such numbers are given with examples. We provide evidence that the quantity prime numbers unlimited and write into the table of prime numbers using Eratosthenes' method. Evidence will be given to determine whether a number is prime or composite.

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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 composite. To understand the concept of composite numbers, you must 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 numbers, 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, having more than two positive divisors.

Any number that is 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. Let's give a 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 only divisible by themselves and 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 is decomposed into 37 and 181. Note that the concepts of prime numbers and coprime 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 numbers reach sizes of 10000 or 1000000000, then you should think about using the Sieve of Eratosthenes.

Let's consider the theorem that explains the last statement.

Theorem 1

The smallest positive divisor other than 1 of a natural number greater than one is a prime number.

Evidence 1

Let us assume that a is a natural number that is greater than 1, b is the smallest non-one divisor of a. It is necessary to prove that b is a prime number using the method of contradiction.

Let's assume that 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 was completed.

From the condition it is clear that a is divided by b, b is divided by b 1, which means that the concept of divisibility is expressed as follows: a = b q and b = b 1 · q 1 , from where 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 for the number a. Inequality 1< b 1 < b Not corresponds, because we find that b is the smallest positive and non-1 divisor of a.

Theorem 2

There are an infinite number of prime numbers.

Evidence 2

Presumably we take a finite number of natural numbers n and denote them as p 1, p 2, …, p n. Let's consider the option of finding a prime number different from those indicated.

Let us take into consideration the number p, which is equal to p 1, p 2, ..., p n + 1. It is not equal to each of the numbers corresponding to prime numbers of the form p 1, p 2, ..., p n. The number p is prime. Then the theorem is considered to be proven. If it is composite, then you need to take the notation p n + 1 and show that the divisor does not coincide 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 find that it would be divisible by pn + 1. Note that the expression p n + 1 dividing the number p equals the sum p 1, p 2, ..., p n + 1. We obtain that the expression p n + 1 The second term of this sum, which equals 1, must be divided, but this is impossible.

It can be seen that any prime number can be found among any number of given prime numbers. It follows that there are infinitely many prime numbers.

Since there are a lot of prime numbers, the tables are limited to the numbers 100, 1000, 10000, and so on.

When compiling a table of prime numbers, you should take into account that such a task requires sequential checking 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 into the table.

Let's look at it step by step.

If you start with the number 2, then it has only 2 divisors: 2 and 1, which means it can be entered into the table. Same with the number 3. The number 4 is composite; it must be decomposed into 2 and 2. The number 5 is prime, which means it can be recorded in the table. Do this until the number 100.

This method inconvenient and long. You can create a table, but you will 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 look at the example tables below. To begin with, the numbers 2, 3, 4, ..., 50 are written down.

Now you need to cross out all the numbers that are multiples of 2. Perform sequential strikethroughs. We get a table like:

We move on to crossing out numbers that are multiples of 5. We get:

Cross out numbers that are multiples of 7, 11. Ultimately the table looks like

Let's move on 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.

Evidence 3

Must be designated b least divisor composite number a. There is an integer q, where a = b · q, and we have that b ≤ q. Inequalities of the form are unacceptable 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.

From the proven theorem it is clear that crossing out 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 begins with 4, and multiples of 3 with 9, and so on until 100.

Compiling such a table using Eratosthenes’ theorem suggests that when all composite numbers are crossed out, prime numbers will remain 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 the composite numbers that do not have a value greater value root of 50. Searching for numbers is done by crossing out.

Before solving, you need 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 the number 898989898989898989 is composite.

Solution

The sum of the digits of a given number is 9 8 + 9 9 = 9 17. This means that the number 9 · 17 is divisible by 9, based on the divisibility test 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 actions should be taken. The most suitable way is to enumerate numbers. During the process, you can find prime and composite numbers. That is, the numbers should not exceed a in value. That is, the number a must be decomposed into prime factors. if this is satisfied, then the number a can be considered prime.

Example 2

Determine the composite or prime number 11723.

Solution

Now you need to find all the 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 number 200 .

For a more accurate estimate of the number 11723, you need to write the expression 108 2 = 11 664, and 109 2 = 11 881 , That 108 2 < 11 723 < 109 2 . It follows that 11723< 109 . Видно, что любое число, которое меньше 109 считается делителем для заданного числа.

When expanding, we find 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 entire process can be depicted as 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 represent the division as a column:

It follows that 11723 is a composite number, because in addition to itself and 1 it has a divisor of 19.

Answer: 11723 is a composite number.

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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 believe 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. (sensible remark from Olga 09/21/12) We, however, remember that prime numbers are sometimes entered like this: " A prime number is a number that is divisible by one and itself."In this case, one is obviously a prime number.

Table of prime numbers from 2 to 1000. Table of prime numbers from 2 to 1000 is highlighted in grey.

Table of prime numbers from 2 to 1000.
The table of prime numbers from 2 to 1000 is highlighted in grey.

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

Table of prime numbers from 1000 to 10,000.

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 the sign :)

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In this article we will explore prime and composite numbers. First, we will give definitions of prime and composite numbers, and also give examples. After this we will prove that there are infinitely many prime numbers. Next, we will write down a table of prime numbers, and consider methods for compiling a table of prime numbers, paying particular attention to the method called the sieve of Eratosthenes. In conclusion, we highlight the main points that need to be taken into account when proving that given number is simple or compound.

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Prime and Composite Numbers - Definitions and Examples

The concepts of prime numbers and composite numbers refer to numbers 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 understanding of what divisors and multiples are.

Definition.

Prime numbers are integers, large units, that have only two positive divisors, namely themselves and 1.

Definition.

Composite numbers are integers, large ones, that have at least three positive divisors.

Separately, we note that the number 1 does not apply to either prime or composite numbers. 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 one has only one positive divisor, we can give other formulations of the stated definitions of prime and composite numbers.

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 or a composite number. In other words, there is not a single integer that is neither prime nor composite. This follows from the property of divisibility, which states 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 composite.

Let's give examples of prime and composite numbers.

Examples of composite numbers include 6, 63, 121, and 6,697. This statement also needs clarification. The number 6, in addition to positive divisors 1 and 6, also has divisors 2 and 3, since 6 = 2 3, therefore 6 is truly a composite number. Positive factors of 63 are the numbers 1, 3, 7, 9, 21 and 63. The number 121 is equal to the product 11·11, so its positive divisors are 1, 11 and 121. And the number 6,697 is composite, since its positive divisors, in addition to 1 and 6,697, are also the numbers 37 and 181.

In conclusion of this point, I would also like to draw attention to the fact that prime numbers and coprime numbers are far from the same thing.

Prime numbers table

Prime numbers, for the convenience of their further use, are recorded in a table called a table of prime numbers. Below is prime numbers table up to 1,000.

Arises logical question: “Why did we fill out the table of prime numbers only up to 1,000, isn’t it possible to make a table of all the prime numbers that exist”?

Let's answer the first part of this question first. For most problems that require the use of prime numbers, prime numbers within a thousand will be sufficient. In other cases, most likely, you will have to resort to some special solutions. Although we can certainly create a table of prime numbers up to an arbitrarily large finite positive integer, be it 10,000 or 1,000,000,000, in the next paragraph we will talk about methods for creating tables of prime numbers, in particular, we will look at a method called.

Now let's look at the possibility (or rather, the impossibility) of compiling a table of all existing prime numbers. We cannot make a table of all the prime numbers because there are infinitely many prime numbers. The last statement is a theorem that we will prove after the following auxiliary theorem.

Theorem.

The smallest positive divisor other than 1 of a natural number greater than one is a prime number.

Proof.

Let a is a natural number greater than one, and b is the smallest positive divisor of a other than one. Let us prove that b is a prime number by contradiction.

Let's assume that 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 condition 1 must be satisfied

Since the number a is divisible by b according to the condition, and we said that b is divisible by b 1, the concept of divisibility allows us to talk about the existence of integers q and q 1 such that a=b q and b=b 1 q 1 , from where a= b 1 ·(q 1 ·q) . 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 an infinite number of prime numbers.

Proof.

Let's assume that this is not the case. That is, suppose that there are only n prime numbers, and these prime numbers are p 1, p 2, ..., p n. Let us show that we can always find a prime number different from those indicated.

Consider the number p equal to p 1 ·p 2 ·…·p n +1. It is clear that this number is different from each of the prime numbers p 1, p 2, ..., p n. If the number p is prime, then the theorem is proven. If this number is composite, then by virtue of the previous theorem there is a prime divisor of this number (we denote it p n+1). Let us show that this divisor does not coincide with any of the numbers p 1, p 2, ..., p n.

If this were not so, then, according to the properties of divisibility, the product p 1 ·p 2 ·…·p n would be divided 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. It follows that p n+1 must divide the second term of this sum, which is equal to one, but this is impossible.

Thus, it has been proven that a new prime number can always be found that is not included among any number of predetermined prime numbers. Therefore, there are infinitely many prime numbers.

So, due to the fact that there are an infinite number of prime numbers, when compiling tables of prime numbers, you always limit yourself from above to some number, usually 100, 1,000, 10,000, etc.

Sieve of Eratosthenes

Now we will discuss ways to create 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 from 2 and ending with 100, for the presence of a positive divisor that is greater than 1 and less than the number being tested (from the properties of divisibility we know that the absolute value of the divisor does not exceed the absolute value of the dividend, non-zero). If such a divisor is not found, then the number being tested is prime, and it is entered into the prime numbers table. If such a divisor is found, then the number being tested is composite; it is NOT entered in the table of prime numbers. After this, the transition occurs 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 simple, 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 the number 2. But 3 is not divisible by 2, therefore, 3 is a prime number, and it also needs to be included in the table of prime numbers. Let's move on to number 4. Its positive divisors other than 1 and 4 can be the numbers 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 included in the table of prime numbers. Please note that 4 is the smallest composite number. Let's move on to number 5. We check whether at least one of the numbers 2, 3, 4 is its divisor. Since 5 is not divisible by 2, 3, or 4, then it is prime, and it must be written down 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 prime numbers is far from ideal. One way or another, he has a 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 create a table of prime numbers, called. The word “sieve” present in the name is not accidental, since the actions of this method help, as it were, to “sift” whole numbers and large units through the sieve of Eratosthenes in order to separate simple ones from composite ones.

Let's show Eratosthenes' sieve in action when compiling a table of prime numbers up to 50.

First, write down the numbers 2, 3, 4, ..., 50 in order.

The first number written, 2, is prime. Now, from number 2, we sequentially move to the right by two numbers and cross out these numbers until we reach the end of the table of numbers being compiled. This will cross out all numbers that are multiples of two.

The first number following 2 that is not crossed out is 3. This number is prime. Now, from number 3, we consistently move to the right by three numbers (taking into account the already crossed out numbers) and cross them out. This will cross out all numbers that are multiples of three.

The first number following 3 that is not crossed out is 5. This number is prime. Now from the number 5 we consistently move to the right by 5 numbers (we also take into account the numbers crossed out earlier) and cross them out. This will cross out all numbers that are multiples of five.

Next, we cross out numbers that are multiples of 7, then multiples of 11, and so on. The process ends when there are no more numbers to cross off. Below is the completed table of prime numbers up to 50, obtained using the sieve of Eratosthenes. All uncrossed numbers are prime, and all crossed out numbers are composite.

Let's also 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 smallest positive divisor of a composite number a that is different from one does not exceed , where is from a .

Proof.

Let us denote by the letter b the smallest divisor of a composite number a that is different from one (the number b is prime, as follows from the theorem proven 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 (for b>q the condition that b is the least divisor of a is violated, since q is also a divisor of the number a due to the equality a=q·b ). By multiplying both sides of the inequality by a positive and an integer greater than one (we are allowed to do this), we obtain , from which and .

What does the proven theorem give us regarding the sieve of Eratosthenes?

Firstly, crossing out 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 begin with the number 4, multiples of three with the number 9, multiples of five with the number 25, and so on.

Secondly, compiling 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 . In our example, n=50 (since we are making a table of prime numbers up to 50) and, therefore, the sieve of Eratosthenes should eliminate all composite numbers that are multiples of the prime numbers 2, 3, 5 and 7 that do not exceed the arithmetic square root of 50. That is, we no longer need to search for 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 general, this task is far from simple, especially for numbers whose writing 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.

Of course, you can try to use divisibility tests to prove that a given number is composite. If, for example, some test of divisibility shows that a given number is divisible by some positive integer greater than one, then the original number is composite.

Example.

Prove that 898,989,898,989,898,989 is a composite number.

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 divisibility by 9 we can say that the original number is also divisible by 9. Therefore, it is composite.

A significant drawback of this approach is that the divisibility criteria do not allow one to prove the primeness of a number. Therefore, when testing a number to see whether it is prime or composite, you need to proceed differently.

The most logical approach is to try all possible divisors of a given number. If none of the possible divisors is a true divisor of a given number, then this number will be prime, otherwise it will be composite. From the theorems proved in the previous paragraph, it follows that divisors of a given number a must be sought among prime numbers not exceeding . Thus, a given number a can be sequentially divided by prime numbers (which are conveniently taken 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 up to what prime number the divisors of the number 11,723 can be. To do this, let's evaluate.

It's pretty obvious that , since 200 2 =40,000, and 11,723<40 000 (при необходимости смотрите статью comparison of numbers). Thus, the possible prime factors of 11,723 are less than 200. This already makes our task much easier. If we didn’t know this, then we would have to go through all the prime numbers not up to 200, but up to the number 11,723.

If desired, you can evaluate more accurately. Since 108 2 =11,664, and 109 2 =11,881, then 108 2<11 723<109 2 , следовательно, . Thus, any of the prime numbers less than 109 is potentially a prime factor 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 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 say right away that 11,723