ExplanationHere is a table of the first 225 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 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 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 The last 10 prime numbers discovery records were held by them. Finding a new prime number is like searching for a needle in a haystack. High-speed super computer are need to try out different numbers and testing them whether they are prime or not. There are few computer scientists who developed programs to accelerate the process of discovering prime numbers. For instance, the “Prime finder” program developed by Slowinski and Gage is being used as a qualification test for a newly developed super computer. This programme involves squiring a number repeatedly thousand of times and in fact is a “torture test” for a computer that has to do the processing continuously for hours. Number of people around the world are participating in the “Great Internet Mersenne Prime Search” to find the next Mersenne Prime. By definition, prime numbers are integers greater than one with no positive divisors besides one and itself. Negative numbers are excluded. This can be explained as follows: Suppose a and –a divide b and we treat them essentially the same divisor. This happens because –1 divides 1, which in turn divides every other number. Numbers that divide one are called units”. Two numbers a and b for which a is a unit times b are called associates. So the divisors a and –a of b are associates. Looking in this way negative integers can be prime. In fact the integer –p is a prime whenever p is a prime number. However, since they are associates, we readily do not have any new primes. For example, complex numbers such as a + bi where a and b are integers and ‘i’ is square root of minus one. Here there are four units, integers that divide, 1, and -1, i and –i. Therefore, each prime has four associates. It is possible to create a system in which each prime has infinitely many associates. The number 1 is excluded from the primes list, since it is a unit, the building block of positive integers. It is the only multiplicative identity, since 1.a=a.1=1. An integer “greater than one” is a called a Prime number if it is only positive divisors (factors) are one and itself. So, by definition the number one excluded from the list of prime numbers. HOW TO FIND A PRIME: The chance of random integer x being a prime is about 1/log x. This is so because, about x/log x of the x positive integers less than or equal to x are prime, the probability of one of them being prime is about 1/log x. Suppose one wants to find a 1000 digit prime number. Choosing 1000 digit integers x, one must test about log (101000) of them or about 2302 integers before finding a prime. Obviously if one used odd integers one could multiplex this estimate by 1/21 and if one chooses integers not divisible by 3, then he multiplies by 2/3 and so on. APPLICATIONS: · there are few applications the prime numbers offer. One has been already mentioned, the quality assurance test of a super computer. · Through the work of finding prime numbers people are learning new techniques for speeding up certain mathematical operations in developing software programs. · Through the prime number a new perfect number can also be generated. A perfect number is equal to the sum of its factors. For example 6 is a perfect number because its factors 1,2,3 when added together equal to 6. Mathematicians do not know how many perfect numbers exist. The 34th perfect number discovered has 757,263 digits. · Prime numbers find applications in cryptography and computer system security.