Wednesday, 14 September 2022

A Special Class of Semiprimes

Looking at the factors of my diurnal age today, I was immediately struck by the fact that each factor was a permutation of the digits of the other. To be specific:

26827 = 139 * 193

I immediately speculated as to how common such an occurrence was. To begin my investigation, I had to exclude square semiprimes such as 121 because they qualify trivially. Here is a permalink to the algorithm that I developed to find all such semiprimes up to one million. Below are the semiprimes along with their factorisation:

403 = 13 * 31

1207 = 17 * 71

2701 = 37 * 73

7663 = 79 * 97

14803 = 113 * 131

23701 = 137 * 173

26827 = 139 * 193

34417 = 127 * 271

35143 = 113 * 311

35263 = 179 * 197

40741 = 131 * 311

43429 = 137 * 317

54841 = 173 * 317

62431 = 149 * 419

70027 = 239 * 293

73159 = 149 * 491

75007 = 107 * 701

89647 = 157 * 571

99919 = 163 * 613

101461 = 241 * 421

102853 = 163 * 631

103039 = 167 * 617

103603 = 313 * 331

117907 = 157 * 751

125701 = 337 * 373

127087 = 167 * 761

128701 = 179 * 719

130771 = 251 * 521

140209 = 149 * 941

141643 = 197 * 719

146791 = 181 * 811

150463 = 379 * 397

153211 = 349 * 439

173809 = 179 * 971

174001 = 191 * 911

182881 = 199 * 919

191287 = 197 * 971

197209 = 199 * 991

201379 = 277 * 727

205729 = 419 * 491

212887 = 359 * 593

230701 = 281 * 821

232909 = 283 * 823

246991 = 367 * 673

247021 = 337 * 733

249979 = 457 * 547

257821 = 347 * 743

273409 = 373 * 733

280081 = 379 * 739

293383 = 397 * 739

295501 = 461 * 641

297709 = 463 * 643

302149 = 467 * 647

326371 = 389 * 839

342127 = 359 * 953

355123 = 379 * 937

367639 = 563 * 653

371989 = 397 * 937

374971 = 569 * 659

382387 = 389 * 983

386803 = 613 * 631

394279 = 419 * 941

427729 = 619 * 691

428821 = 571 * 751

436789 = 577 * 757

453613 = 479 * 947

462031 = 491 * 941

469537 = 617 * 761

503059 = 587 * 857

565129 = 593 * 953

589429 = 683 * 863

643063 = 709 * 907

690199 = 787 * 877

692443 = 739 * 937

698149 = 719 * 971

743623 = 769 * 967

778669 = 797 * 977

824737 = 839 * 983

910729 = 919 * 991

There are 79 semiprimes in total. Here is the list (I'm surprised this sequence hasn't made it into the OEIS but it hasn't and I've no intention of submitting it):

403, 1207, 2701, 7663, 14803, 23701, 26827, 34417, 35143, 35263, 40741, 43429, 54841, 62431, 70027, 73159, 75007, 89647, 99919, 101461, 102853, 103039, 103603, 117907, 125701, 127087, 128701, 130771, 140209, 141643, 146791, 150463, 153211, 173809, 174001, 182881, 191287, 197209, 201379, 205729, 212887, 230701, 232909, 246991, 247021, 249979, 257821, 273409, 280081, 293383, 295501, 297709, 302149, 326371, 342127, 355123, 367639, 371989, 374971, 382387, 386803, 394279, 427729, 428821, 436789, 453613, 462031, 469537, 503059, 565129, 589429, 643063, 690199, 692443, 698149, 743623, 778669, 824737, 910729

As can be seen, 26827 is only the seventh such number. I may well be dead before I see the next such semiprime (34417) that will represent my diurnal age on June 26th 2043. If I'm still here I would have passed my 94th birthday.

Another approach to finding these semiprimes would be to test all the primes in a given range, find what permutations of the digits produce primes and multiply the two together. However, this doesn't make for a very efficient algorithm. When I tried it on SageMathCell, the program timed out so the algorithm linked to in the permalink is far more efficient, producing a speedy output.

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