Sequence of Composite Mersenne Numbers

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Theorem

The sequence of Mersenne numbers which are composite begins:

$2047, 8 \, 388 \, 607, 536 \, 870 \, 911, 137 \, 438 \, 953 \, 471, 2 \, 199 \, 023 \, 255 \, 551,\ldots$

This sequence is A065341 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The sequence of corresponding indices $p$ such that $2^p - 1$ is composite begins:

$11, 23, 29, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, \ldots$

This sequence is A054723 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The sequence of corresponding integers $n$ such that the $n$th prime number $p \left({n}\right)$ is such that $2^{p \left({n}\right)} - 1$ is composite begins:

$5, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, \ldots$

This sequence is A135980 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Examples of Composite Mersenne Numbers

Mersenne Number $M_{67}$

$M_{67}$ (that is, $2^{67} - 1$) is a composite number:

\(\displaystyle 2^{67} - 1\) \(=\) \(\displaystyle 147 \, 573 \, 952 \, 589 \, 676 \, 412 \, 927\)
\(\displaystyle \) \(=\) \(\displaystyle 193 \, 707 \, 721 \times 761 \, 838 \, 257 \, 287\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 1 \, 445 \, 580 \times 67 + 1}\right) \times \left({2 \times 5 \, 685 \, 360 \, 129 \times 67 + 1}\right)\)


Mersenne Number $M_{157}$

$M_{157}$ (that is, $2^{157} - 1$) is a composite number:

\(\displaystyle 2^{157} - 1\) \(=\) \(\displaystyle 182 \, 687 \, 704 \, 666 \, 362 \, 864 \, 775 \, 460 \, 604 \, 089 \, 535 \, 377 \, 456 \, 991 \, 567 \, 871\)
\(\displaystyle \) \(=\) \(\displaystyle 852 \, 133 \, 201 \times 60 \, 726 \, 444 \, 167 \times 1 \, 654 \, 058 \, 017 \, 289 \times 2 \, 134 \, 387 \, 368 \, 610 \, 417\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 2 \, 713 \, 800 \times 157 + 1}\right) \times \left({2 \times 193 \, 396 \, 319 \times 157 + 1}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({2 \times 5 \, 267 \, 700 \, 692 \times 157 + 1}\right) \times \left({2 \times 6 \, 797 \, 412 \, 001 \, 944 \times 157 + 1}\right)\)


Mersenne Number $M_{167}$

$M_{167}$ (that is, $2^{167} - 1$) is a composite number:

\(\displaystyle 2^{167} - 1\) \(=\) \(\displaystyle 187 \, 072 \, 209 \, 578 \, 355 \, 573 \, 530 \, 071 \, 658 \, 587 \, 684 \, 226 \, 515 \, 959 \, 365 \, 500 \, 927\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \, 349 \, 023 \times 79 \, 638 \, 304 \, 766 \, 856 \, 507 \, 377 \, 778 \, 616 \, 296 \, 087 \, 448 \, 490 \, 695 \, 649\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 7033 \times 167 + 1}\right) \times \left({2 \times 238 \, 438 \, 038 \, 224 \, 121 \, 279 \, 574 \, 187 \, 473 \, 940 \, 381 \, 582 \, 307 \, 472 \times 167 + 1}\right)\)


Mersenne Number $M_{193}$

$M_{193}$ (that is, $2^{193} - 1$) is a composite number:

\(\displaystyle 2^{193} - 1\) \(=\) \(\displaystyle 12 \, 554 \, 203 \, 470 \, 773 \, 361 \, 527 \, 671 \, 578 \, 846 \, 415 \, 332 \, 832 \, 204 \, 710 \, 888 \, 928 \, 069 \, 025 \, 791\)
\(\displaystyle \) \(=\) \(\displaystyle 13 \, 821 \, 503 \times 61 \, 654 \, 440 \, 233 \, 248 \, 340 \, 616 \, 559 \times 14 \, 732 \, 265 \, 321 \, 145 \, 317 \, 331 \, 353 \, 282 \, 383\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 35 \, 807 \times 193 + 1}\right) \times \left({2 \times 159 \, 726 \, 529 \, 101 \, 679 \, 638 \, 903 \times 193 + 1}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({2 \times 38 \, 166 \, 490 \, 469 \, 288 \, 386 \, 868 \, 790 \, 887 \times 193 + 1}\right)\)


Mersenne Number $M_{199}$

$M_{199}$ (that is, $2^{199} - 1$) is a composite number:

\(\displaystyle 2^{199} - 1\) \(=\) \(\displaystyle 803 \, 469 \, 022 \, 129 \, 495 \, 137 \, 770 \, 981 \, 046 \, 170 \, 581 \, 301 \, 261 \, 101 \, 496 \, 891 \, 396 \, 417 \, 650 \, 687\)
\(\displaystyle \) \(=\) \(\displaystyle 164 \, 504 \, 919 \, 713 \times 4 \, 884 \, 164 \, 093 \, 883 \, 941 \, 177 \, 660 \, 049 \, 098 \, 586 \, 324 \, 302 \, 977 \, 543 \, 600 \, 799\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 413 \, 328 \, 944 \times 199 + 1}\right) \times \left({2 \times 12 \, 271 \, 769 \, 080 \, 110 \, 404 \, 968 \, 995 \, 098 \, 237 \, 654 \, 081 \, 163 \, 260 \, 159 \, 801 \times 199 + 1}\right)\)


Mersenne Number $M_{227}$

$M_{227}$ (that is, $2^{227} - 1$) is a composite number:

\(\displaystyle 2^{227} - 1\) \(=\) \(\displaystyle 215 \, 679 \, 573 \, 337 \, 205 \, 118 \, 357 \, 336 \, 120 \, 696 \, 157 \, 045 \, 389 \, 097 \, 155 \, 380 \, 324 \, 579 \, 848 \, 828 \, 881 \, 993 \, 727\)
\(\displaystyle \) \(=\) \(\displaystyle 26 \, 986 \, 333 \, 437 \, 777 \, 017 \times 7 \, 992 \, 177 \, 738 \, 205 \, 979 \, 626 \, 491 \, 506 \, 950 \, 867 \, 720 \, 953 \, 545 \, 660 \, 121 \, 688 \, 631\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 59 \, 441 \, 263 \, 078 \, 804 \times 227 + 1}\right) \times \left({2 \times 17 \, 603 \, 915 \, 722 \, 920 \, 659 \, 970 \, 245 \, 610 \, 023 \, 937 \, 711 \, 351 \, 422 \, 158 \, 858 \, 345 \times 227 + 1}\right)\)


Mersenne Number $M_{229}$

$M_{229}$ (that is, $2^{229} - 1$) is a composite number:

\(\displaystyle 2^{229} - 1\) \(=\) \(\displaystyle 862 \, 718 \, 293 \, 348 \, 820 \, 473 \, 429 \, 344 \, 482 \, 784 \, 628 \, 181 \, 556 \, 388 \, 621 \, 521 \, 298 \, 319 \, 395 \, 315 \, 527 \, 974 \, 911\)
\(\displaystyle \) \(=\) \(\displaystyle 1 \, 504 \, 073 \times 20 \, 492 \, 753 \times 59 \, 833 \, 457 \, 464 \, 970 \, 183 \times 467 \, 795 \, 120 \, 187 \, 583 \, 723 \, 534 \, 280 \, 000 \, 348 \, 743 \, 236 \, 593\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 3284 \times 229 + 1}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({2 \times 44744 \times 229 + 1}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({2 \times 130 \, 640 \, 736 \, 823 \, 079 \times 229 + 1}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({2 \times 1 \, 021 \, 386 \, 725 \, 300 \, 401 \, 143 \, 087 \, 947 \, 599 \, 014 \, 723 \, 224 \times 229 + 1}\right)\)


Mersenne Number $M_{257}$

$M_{257}$ (that is, $2^{257} - 1$) is a composite number:

\(\displaystyle 2^{257} - 1\) \(=\) \(\displaystyle 231 \, 584 \, 178 \, 474 \, 632 \, 390 \, 847 \, 141 \, 970 \, 017 \, 375 \, 815 \, 706 \, 539 \, 969 \, 331 \, 281 \, 128 \, 078 \, 915 \, 168 \, 015 \, 826 \, 259 \, 279 \, 871\)
\(\displaystyle \) \(=\) \(\displaystyle 535 \, 006 \, 138 \, 814 \, 359 \times 1 \, 155 \, 685 \, 395 \, 246 \, 619 \, 182 \, 673 \, 033 \times 374 \, 550 \, 598 \, 501 \, 810 \, 936 \, 581 \, 776 \, 630 \, 096 \, 313 \, 181 \, 393\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {2 \times 1 \, 040 \, 867 \, 974 \, 347 \times 257 + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \paren {2 \times 2 \, 248 \, 415 \, 165 \, 849 \, 453 \, 662 \, 788 \times 257 + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \paren {2 \times 728 \, 697 \, 662 \, 454 \, 885 \, 090 \, 626 \, 024 \, 572 \, 171 \, 815 \, 528 \times 257 + 1}\)


Proof

Established by inspecting the sequence of Mersenne numbers:

$3, 7, 31, 127, 2047, 8191, 131 \, 071, 524 \, 287, 8 \, 388 \, 607, 536 \, 870 \, 911, 2 \, 147 \, 483 \, 647, \ldots$

and removing from it the sequence of Mersenne primes:

$3, 7, 31, 127, 8191, 131 \, 071, 524 \, 287, 2 \, 147 \, 483 \, 647, \ldots$

$\blacksquare$


Sources