A prime is an integer that has exactly 2 positive divisors.
This excludes 1 which is not prime, and which the definition of prime as "number divisible by itself and one" allows. 1 is not prime.

This also means -1 is not prime, since -1 does not have exactly two positive divisors. And, -3 is prime.

A prime number is a whole number that is divisible only by one and itself. 3 can is divisible by 1 and 3, and is prime. 6 is divisible by 1, 2, 3, and 6, and is not prime. 1 is not counted as prime, making 2 the first prime number, and the only even prime number.

Below is a brief list of prime numbers and their applications, as noded on E2. /msg me with any additions*.


up^

Prime numbers were first studied extensively by ancient Greek mathematicians. The Pythagorean school (500 BC to 300 BC) where interested in the numerological properties. By Euclid's time (300 BC), several important results about primes had been proved. Euclid went on to prove that there are an infinite number of prime numbers. This proof is the first known one using the method of reduction to absurd. Euclid also proved the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way. In 200 BC, Eratosthanes devised an algorithm for calculating primes called the Sieve of Eratosthanes.

There is a great gap in the history of prime numbers during the Dark Ages.

The next major development was made by Fermat in the 17th Century. Fermat proved a speculation of Albert Girard that every prime of the form 4n + 1 can be written in a unique way as the sum of two squares, and any number could be written as the sum of four squares. In his corespondance with Mersenne, Fermat conjectured that 2n + 1 was always prime if n was a power of 2. He had verified this for n = 1, 2, 4, 8, and 16 but was did not know 232+1 was prime or not. 100 years later Euler showed that 232 + 1 was 4294967297 and is divisible by 641, hence not prime.

The next major step in prime number theory came from a monk named Mersenne. Mersenne studied many numbers of the form 2n - 1 and a class of numbers named Mersenne numbers were named after him. These numbers attracted attention because it was easy to show that unless n was prime the number must be composite. Not all of these numbers are prime, though for many years (and again today) they provided the largest prime numbers. For 200 years M19 was the largest known prime until Euler proved that M31 is prime. This established the record for another 100 years until M127 is prime until the age of the computer. In 1952 the Mersenne numbers M521, M607, M1279, M2203 and M2281 were proved to be prime with the aid of a computer.

Euler's work extended that of Fermat's working with amicable numbers and stated what is known today as the Law of Quadratic Reciprocity. Euler also worked with sequences of primes and divergent series.
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...
is a divergent series, though even the most powerful computers today can only sum it to about 4.

Gauss studied the density of prime numbers along the integers. In the first 100 integers there are 9 primes, while the next 100 only has 2. Gauss showed that on a large scale, the distribution is very regular. Gauss once told a friend that whenever he had a spare 15 minutes he would spend it counting numbers in a 'chiliad' (a rage of 1000 numbers). It was estimated that by the end of his life he had counted all the primes up to about 3 million.

Today the counting of prime numbers continues, the most well known is GIMPS, The Great Internet Mersenne Prime Search.

The reason for not allowing 1 as prime is to keep the fundamental theorem of arithmetic. If you could say 24=2*2*2*3=1*1*1*1*2*2*2*3 were both decompositions of 24 into prime factors, the decomposition would not be unique.

Pakaran is incorrect regarding "prime" complex numbers. The integer ring of the algebraic numbers is the ring of gaussian integers -- numbers of the form a+bi with a,b both integers. In that ring, as Noether's excellent gaussian integers writeup shows, an integer p that is prime as an integer is prime iff it is not of the form 4k+1. When p=4k+1, Noether shows it is possible to write p = a2+b2 = (a+bi)(a-bi), so p is not prime. But since the norm squared (absolute value) of a±bi is p, and since the norm squared of any gaussian integer is integer and multiplicative, it follows that a±bi itself is a prime gaussian integer.

Thus we can classify all prime gaussian integers.

Um, small writeup here, but all prime numbers other than 2 and 3 are one more or one less than a multiple of 6.

So if you're looking for prime numbers, or testing to see if a number is prime, knowing this cuts out 2/3 of the numbers for you.


Good point, Gritchka. ok, to show how this is useful:
finding the prime factors of a number:
(in C, to make my life easier)

void PrintFactors(int number) {
   int a,b,limit;

   while ((number%2)==0) printf("2*",number/=2);
   while ((number%3)==0) printf("3*",number/=3);

   a = 6-1;
   b = 6+1;

   limit = isqrt(number); /* integer square root */

   while (a < limit) {

      while ((number%a)==0) printf("%d*",a,number/=a);
      while ((number%b)==0) printf("%d*",b,number/=b);

      a += 6;  b += 6;
   }

   printf("%d",number);
}

(forgive my cheating with extra params for printf)
This should take about 2/3 as long as something that checked every odd number, or 1/3 as long as something that checked every integer.

Or if you're trying to do the Sieve of Eratosthanes for some reason, you can save time and memory by not representing other numbers at all. Have a table of all numbers that are 6n+1 and another for 6n-1.

The following was in response to rabidcow's first two paragraphs only; it is not a criticism of the C program added afterwards.

Um, I don't think that is as helpful as it would like to be. How do you know something is a multiple of 6? It has to be (a) a multiple of 2; (b) a multiple of 3.

To check (a) look at the last digit; to check (b) add up all the digits (repeatedly if necessary) to see whether their sum is divisible by 3. Both simple operations.

But any number is precisely one of

  1. a multiple of 6
  2. 1 more than a multiple of 6
  3. 2 more than a multiple of 6
  4. 3 more than a multiple of 6
  5. 4 more than a multiple of 6
  6. 5 more than a multiple of 6
These can be checked with
  1. either (a) or (b), but (a) is easier
  2. neither
  3. (a)
  4. (b)
  5. (a)
  6. neither
Being 5 more than is the same as being 1 less than (modulo 6), so this leaves the two possibilities rabidcow mentioned. But in no case do you have to do the (relatively) hard work of checking both (a) and (b). The o(1) algorithm (a) eliminates half and the o(n) algorithm (b) eliminates a further one-sixth.

If it's still a candidate after this you have to do work, but its passing these tests has incidentally told you that it's a multiple of 6, plus or minus 1, a converse of the previous write-up's idea.

Here is the proof of two statements above. Just in case someone is interested.

It is easy to prove that the set of all prime numbers is infinite(more precisely - there is no greatest prime number). Here is how.

Let the set be finite and consist of {n1,n2,...nk} Where nk is the largest prime number.
Then the number n1*n2*n3.....nk+1 does not belong to the above set. Also it is not divisible by either n1 or n2 or n3 ... or nk. So either it is prime or it has a prime factor larger than nk. In either case, we have found a prime number greater than nk. This contradicts the assumption that nk was the largest prime number. So the assumption is false and there is no largest prime.

The second statement about all primes except for 2 and 3 being either 1 less or one more than 6. Here is why this is true.

Any positive natural number may be written in the form 6n+k. Where n is a natural number and k is one of {0,1,2,3,4,5}. A prime number cannot have k = 0 or k = 2 or k=4 because in all these cases, the number would be divisible by 2. It cannot have k = 3 because in this case it would be divisible by 3 as 6n+3 = 3*(2n+1). Thus it must be of the form 6n+1 or 6n+5=6(n+1)-1. So it must be one more or one less than a multiple of 6.

If you take any two prime numbers above 3 and square them, the difference between them is always divisible by 24.

This is an observation i made one night while trying to go to sleep. I later checked my theory with all of the prime numbers from 5 to 101 and it worked. I'm afraid i don't know enough math to write this as a proof. I was looking for some kind of connection between the primes, and squared them as a means of finding that connection. I soon noticed that the differences between the squared primes was always divisible by 8, and later, 3 as well (thus 24).

Definition: A number n in the natural numbers is composite iff there exists a number 1 < k < n in the naturals such that k|n. (The "|" symbol means "divides".)

From the definition of 'divides', for numbers a,b in the naturals, a|b iff there exists a number c in the integers such that a*c=b.

In other words, n is composite iff there exist naturals 1 < k < n and x such that k * x = n.

The inverse: A number n in the naturals is prime iff for all 1 < k < n and x in the naturals, k*x != n. (!= is "does not equal".)

Extension: 1 < k < n, so k=2,3,4,...,n-1. Consider 2 to be 'already' prime, so all numbers k = 2*g for g in the naturals are not prime. Also, consider 3 to be 'already' prime, so all k=3*h for h in the naturals are not prime. So, ignore k=2,3,4,6,8,9,10,..., i.e. let k=6*u±1 for u in the naturals; let n and x be of this form as well, so n=6*m±1 and x=6*v±1 for m, v in the naturals.

Now the number of tests required for primality are reduced by a factor of 3. To test if n is prime, look for any k,x for which k*x=n, or test the equations 6*m±1=(6*u±1)*(6*v±1) for any solutions. Keep in mind that 6*m+1 is a separate case from 6*m-1, but all of 6*u+1, 6*u-1, 6*v+1 and 6*v-1 must be tested.

Distributing the above multiplication yields
6*m±1=36*u*v±6*u±6*v±1
Some manipulation and mod operations reduce this to:
m=6*u*v±u±v
A little more work yields some insight as to some of the larger implications of this equation. One thing in particular is that, if there are *no* u and v in the naturals to satisfy the above equations, then 6*m+1 and 6*m-1 are prime.


Some prime curios:

1234567891 is prime. In fact, so is 12345678901234567891. Both of these may be checked with primo in a very short amount of time. Another interesting prime: (37)14413. That's right: write '37' 1441 times in a row next to each other, then put a 3 on the end and you have a prime. I have seen some conflicting reports, but currently (2^24,036,583)-1 is the largest prime known to date that I can confirm. In terms of digits, this is roughly 7 million decimal digits.

The primes below 1000 are as follows:

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
A proof of lisapple's unproven statement:

For every prime > 3 (by the above logic), there exists a number n such that said prime is equal to either b or c, where:

b = 6n+1, c = 6n-1.

Therefore:

b^2 = 36n^2+12n+1
c^2 = 36n^2-12n+1
The difference between any two primes >3, therefore, are equal to:

(b1)^2-(b2)^2 = 36n^2 - 36m^2 +12n - 12m = 12 (3n^2 - 3m^2 + n - m).
The value in the parentheses is even for all integer values of m and n.
Therefore, there exists a number L such that (a1)^2 - (a2)^2 = 24L.
And 24 divides 24L.

The other cases are solvable by similar logic.
Q.E.D.

Not nearly enough prime!

All the primes below 20000:

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 10007 10009 10037 10039 10061 10067 10069 
10079 10091 10093 10099 10103 10111 10133 10139 10141 10151 10159 10163 
10169 10177 10181 10193 10211 10223 10243 10247 10253 10259 10267 10271 
10273 10289 10301 10303 10313 10321 10331 10333 10337 10343 10357 10369 
10391 10399 10427 10429 10433 10453 10457 10459 10463 10477 10487 10499 
10501 10513 10529 10531 10559 10567 10589 10597 10601 10607 10613 10627 
10631 10639 10651 10657 10663 10667 10687 10691 10709 10711 10723 10729 
10733 10739 10753 10771 10781 10789 10799 10831 10837 10847 10853 10859 
10861 10867 10883 10889 10891 10903 10909 10937 10939 10949 10957 10973 
10979 10987 10993 11003 11027 11047 11057 11059 11069 11071 11083 11087 
11093 11113 11117 11119 11131 11149 11159 11161 11171 11173 11177 11197 
11213 11239 11243 11251 11257 11261 11273 11279 11287 11299 11311 11317 
11321 11329 11351 11353 11369 11383 11393 11399 11411 11423 11437 11443 
11447 11467 11471 11483 11489 11491 11497 11503 11519 11527 11549 11551 
11579 11587 11593 11597 11617 11621 11633 11657 11677 11681 11689 11699 
11701 11717 11719 11731 11743 11777 11779 11783 11789 11801 11807 11813 
11821 11827 11831 11833 11839 11863 11867 11887 11897 11903 11909 11923 
11927 11933 11939 11941 11953 11959 11969 11971 11981 11987 12007 12011 
12037 12041 12043 12049 12071 12073 12097 12101 12107 12109 12113 12119 
12143 12149 12157 12161 12163 12197 12203 12211 12227 12239 12241 12251 
12253 12263 12269 12277 12281 12289 12301 12323 12329 12343 12347 12373 
12377 12379 12391 12401 12409 12413 12421 12433 12437 12451 12457 12473 
12479 12487 12491 12497 12503 12511 12517 12527 12539 12541 12547 12553 
12569 12577 12583 12589 12601 12611 12613 12619 12637 12641 12647 12653 
12659 12671 12689 12697 12703 12713 12721 12739 12743 12757 12763 12781 
12791 12799 12809 12821 12823 12829 12841 12853 12889 12893 12899 12907 
12911 12917 12919 12923 12941 12953 12959 12967 12973 12979 12983 13001 
13003 13007 13009 13033 13037 13043 13049 13063 13093 13099 13103 13109 
13121 13127 13147 13151 13159 13163 13171 13177 13183 13187 13217 13219 
13229 13241 13249 13259 13267 13291 13297 13309 13313 13327 13331 13337 
13339 13367 13381 13397 13399 13411 13417 13421 13441 13451 13457 13463 
13469 13477 13487 13499 13513 13523 13537 13553 13567 13577 13591 13597 
13613 13619 13627 13633 13649 13669 13679 13681 13687 13691 13693 13697 
13709 13711 13721 13723 13729 13751 13757 13759 13763 13781 13789 13799 
13807 13829 13831 13841 13859 13873 13877 13879 13883 13901 13903 13907 
13913 13921 13931 13933 13963 13967 13997 13999 14009 14011 14029 14033 
14051 14057 14071 14081 14083 14087 14107 14143 14149 14153 14159 14173 
14177 14197 14207 14221 14243 14249 14251 14281 14293 14303 14321 14323 
14327 14341 14347 14369 14387 14389 14401 14407 14411 14419 14423 14431 
14437 14447 14449 14461 14479 14489 14503 14519 14533 14537 14543 14549 
14551 14557 14561 14563 14591 14593 14621 14627 14629 14633 14639 14653 
14657 14669 14683 14699 14713 14717 14723 14731 14737 14741 14747 14753 
14759 14767 14771 14779 14783 14797 14813 14821 14827 14831 14843 14851 
14867 14869 14879 14887 14891 14897 14923 14929 14939 14947 14951 14957 
14969 14983 15013 15017 15031 15053 15061 15073 15077 15083 15091 15101 
15107 15121 15131 15137 15139 15149 15161 15173 15187 15193 15199 15217 
15227 15233 15241 15259 15263 15269 15271 15277 15287 15289 15299 15307 
15313 15319 15329 15331 15349 15359 15361 15373 15377 15383 15391 15401 
15413 15427 15439 15443 15451 15461 15467 15473 15493 15497 15511 15527 
15541 15551 15559 15569 15581 15583 15601 15607 15619 15629 15641 15643 
15647 15649 15661 15667 15671 15679 15683 15727 15731 15733 15737 15739 
15749 15761 15767 15773 15787 15791 15797 15803 15809 15817 15823 15859 
15877 15881 15887 15889 15901 15907 15913 15919 15923 15937 15959 15971 
15973 15991 16001 16007 16033 16057 16061 16063 16067 16069 16073 16087 
16091 16097 16103 16111 16127 16139 16141 16183 16187 16189 16193 16217 
16223 16229 16231 16249 16253 16267 16273 16301 16319 16333 16339 16349 
16361 16363 16369 16381 16411 16417 16421 16427 16433 16447 16451 16453 
16477 16481 16487 16493 16519 16529 16547 16553 16561 16567 16573 16603 
16607 16619 16631 16633 16649 16651 16657 16661 16673 16691 16693 16699 
16703 16729 16741 16747 16759 16763 16787 16811 16823 16829 16831 16843 
16871 16879 16883 16889 16901 16903 16921 16927 16931 16937 16943 16963 
16979 16981 16987 16993 17011 17021 17027 17029 17033 17041 17047 17053 
17077 17093 17099 17107 17117 17123 17137 17159 17167 17183 17189 17191 
17203 17207 17209 17231 17239 17257 17291 17293 17299 17317 17321 17327 
17333 17341 17351 17359 17377 17383 17387 17389 17393 17401 17417 17419 
17431 17443 17449 17467 17471 17477 17483 17489 17491 17497 17509 17519 
17539 17551 17569 17573 17579 17581 17597 17599 17609 17623 17627 17657 
17659 17669 17681 17683 17707 17713 17729 17737 17747 17749 17761 17783 
17789 17791 17807 17827 17837 17839 17851 17863 17881 17891 17903 17909 
17911 17921 17923 17929 17939 17957 17959 17971 17977 17981 17987 17989 
18013 18041 18043 18047 18049 18059 18061 18077 18089 18097 18119 18121 
18127 18131 18133 18143 18149 18169 18181 18191 18199 18211 18217 18223 
18229 18233 18251 18253 18257 18269 18287 18289 18301 18307 18311 18313 
18329 18341 18353 18367 18371 18379 18397 18401 18413 18427 18433 18439 
18443 18451 18457 18461 18481 18493 18503 18517 18521 18523 18539 18541 
18553 18583 18587 18593 18617 18637 18661 18671 18679 18691 18701 18713 
18719 18731 18743 18749 18757 18773 18787 18793 18797 18803 18839 18859 
18869 18899 18911 18913 18917 18919 18947 18959 18973 18979 19001 19009 
19013 19031 19037 19051 19069 19073 19079 19081 19087 19121 19139 19141 
19157 19163 19181 19183 19207 19211 19213 19219 19231 19237 19249 19259 
19267 19273 19289 19301 19309 19319 19333 19373 19379 19381 19387 19391 
19403 19417 19421 19423 19427 19429 19433 19441 19447 19457 19463 19469 
19471 19477 19483 19489 19501 19507 19531 19541 19543 19553 19559 19571 
19577 19583 19597 19603 19609 19661 19681 19687 19697 19699 19709 19717 
19727 19739 19751 19753 19759 19763 19777 19793 19801 19813 19819 19841 
19843 19853 19861 19867 19889 19891 19913 19919 19927 19937 19949 19961 
19963 19973 19979 19991 19993 19997 

The following is c++ code for finding whether or not a number is prime:

#include <iostream.h>
/* Program that finds whether a number is prime or not.*/
int main ()
{

int number, notprime, P;
cout << "Enter integer ";
cin >> number; //takes in an integer to test if it's a prime
P=2;
do{ //section for finding whether or not selected integer is a prime
if (number%P==0)
notprime = 1;
else
P++;
}while ((P<=sqrt(number))||(notprime==1));
if (notprime==1)
cout << "Not prime";
else
cout << "Prime";
return (0);
}

Eratosthenes' sieve

One way to find prime numbers up to a given value “x” is to list the positive integers up to and including that number.

Then cross out 1 because 1 is not a prime number

Then take the next integer 2 don’t cross it out but cross out all multiples of it. Repeat this process for all integers up to (and including) the square root of “x”.

(In fact you need only do this for the prime numbers up root “x” but if you are looking for prime numbers you might not know them all and doing all integers is a failsafe method)

As you go further along the list more and more of the numbers will already be crossed out so you will be crossing out fewer and fewer numbers. When you reach root “x” all the numbers you have not yet crossed out are prime.

The question is will there ever reach a point where no matter how big “x” is you won’t have to cross out any more numbers. This would happen is there were a finite number of prime numbers. So are there?

Are there infinite prime numbers?

It is in fact fairly easy to prove that there are infinite prime numbers.

Suppose that there were a finite number of prime numbers.

Then suppose you multiplied them all together.

Then suppose you added one.

The number you would produce is not divisible by any of the other prime numbers because it is one more than multiples of all of them therefore it is either prime or divisible by some other prime number not on the origional list.

You can use this method to generate infinite prime numbers.

e.g.

2*3*5*7*11*13*17*19*23*29*31*37*41*43*47+1 is not prime but it is a multiple of 127-a prime not on the list.

Thanks to 10998521 for pointing this out

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