Prime numbers

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Take a number like 231. It can be factored as 3 x 7 x 11. But the factors 3, 7, and 11 can't be factored any further: they are prime numbers. Primes are to numbers as atoms are to molecules: the basic building blocks from which everything is constructed.

Here are the primes less than 100:

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

There are twenty-five of them. Here are the primes between 10,000 and 10,100:

10007 10009 10037 10039 10061 10067 10069 10079 10091 10093 10099

There are eleven of them.

The primes seem to "thin out" as the numbers get bigger and bigger. This leads us to the question: do we ever run out of primes? Well, it turns out that there are infinitely many primes! This fact was discovered by the Greeks around 300 BC. The proof is found in Euclid's Elements.

Factoring

It is not too difficult to factor small numbers like 123 or 1234567. But factoring larger numbers, like 1020030004000050000060000007, is more difficult. And the effort required rises very rapidly as the number of digits increases. On the other hand, it turns out to be easy, using some number theory, to find very large primes. Here is one:

1000000000000000000000000000000000000000063

It is a 1 followed by forty zeros followed by 63. It is easy to generate large primes like this, but difficult to break large numbers into primes. This is the basis of RSA cryptography.

Question. How do you know that 127 is a prime? What about a larger number like 882389? What about a really large number like

1000000000000000000000000000000000000000063?

References

Integer factoring by Arjen Lenstra

A collection of factoring papers

Some primes. Do you see any patterns?

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