A semisimple number (or a biprime ) is a number that can be represented as a product of two primes .
Examples
The sequence of semisimple numbers starts like this:
- 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, ... (sequence A001358 in OEIS )
The distribution diagram of semisimple numbers on the number axis :
On 07.06.2019 the largest known semisimple number is (2 82589933 - 1) 2 . It is equal to the square of the largest known prime number , which is a prime Mersenne number M 82589933 = 2 82589933 - 1.
The following table lists all semisimple numbers, whose simple divisors do not exceed 47:
| × | 2 | 3 | five | 7 | eleven | 13 | 17 | nineteen | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2 | four | 6 | ten | 14 | 22 | 26 | 34 | 38 | 46 | 58 | 62 | 74 | 82 | 86 | 94 |
| 3 | 6 | 9 | 15 | 21 | 33 | 39 | 51 | 57 | 69 | 87 | 93 | 111 | 123 | 129 | 141 |
| five | ten | 15 | 25 | 35 | 55 | 65 | 85 | 95 | 115 | 145 | 155 | 185 | 205 | 215 | 235 |
| 7 | 14 | 21 | 35 | 49 | 77 | 91 | 119 | 133 | 161 | 203 | 217 | 259 | 287 | 301 | 329 |
| eleven | 22 | 33 | 55 | 77 | 121 | 143 | 187 | 209 | 253 | 319 | 341 | 407 | 451 | 473 | 517 |
| 13 | 26 | 39 | 65 | 91 | 143 | 169 | 221 | 247 | 299 | 377 | 403 | 481 | 533 | 559 | 611 |
| 17 | 34 | 51 | 85 | 119 | 187 | 221 | 289 | 323 | 391 | 493 | 527 | 629 | 697 | 731 | 799 |
| nineteen | 38 | 57 | 95 | 133 | 209 | 247 | 323 | 361 | 437 | 551 | 589 | 703 | 779 | 817 | 893 |
| 23 | 46 | 69 | 115 | 161 | 253 | 299 | 391 | 437 | 529 | 667 | 713 | 851 | 943 | 989 | 1081 |
| 29 | 58 | 87 | 145 | 203 | 319 | 377 | 493 | 551 | 667 | 841 | 899 | 1073 | 1189 | 1247 | 1363 |
| 31 | 62 | 93 | 155 | 217 | 341 | 403 | 527 | 589 | 713 | 899 | 961 | 1147 | 1271 | 1333 | 1457 |
| 37 | 74 | 111 | 185 | 259 | 407 | 481 | 629 | 703 | 851 | 1073 | 1147 | 1369 | 1517 | 1591 | 1739 |
| 41 | 82 | 123 | 205 | 287 | 451 | 533 | 697 | 779 | 943 | 1189 | 1271 | 1517 | 1681 | 1763 | 1927 |
| 43 | 86 | 129 | 215 | 301 | 473 | 559 | 731 | 817 | 989 | 1247 | 1333 | 1591 | 1763 | 1849 | 2021 |
| 47 | 94 | 141 | 235 | 329 | 517 | 611 | 799 | 893 | 1081 | 1363 | 1457 | 1739 | 1927 | 2021 | 2209 |
Properties
- It is proved that every sufficiently large odd natural number can be represented as the sum of three semisimple numbers [1] [2] .
- The square of any prime number is a semisimple number, which is trivial .
- All semisimple numbers except 6 are insufficient .
- If n - 1 and n + 1 are prime numbers twins for some positive integer n, then n 2 - 1 is a semisimple number.