Factors of Mersenne Number M67

Theorem
The Mersenne number $M_{67}$ has the factors:
 * $193 \, 707 \, 721$
 * $761 \, 838 \, 257 \, 287$

Proof
First we calculate $M_{67}$, which is $2^{67} - 1$:


 * $2^2 = 2 \times 2 = 4$

as follows: 256 x 256 -- 1 536 12 800 51 200 -- 65 536 --

as follows: 65 536    x 65 536 ---     393 216    1 966 080   32 768 000  327 680 000 3 932 160 000 - 4 924 967 296 -

as follows: 4 294 967 296          x 4 294 967 296 --           25 769 803 776           386 547 056 640           858 993 459 200        30 064 771 072 000       257 698 037 760 000     3 865 470 566 400 000    17 179 869 184 000 000   386 547 056 640 000 000   858 993 459 200 000 000 17 179 869 184 000 000 000 -- 18 446 744 073 709 551 616 -- 1 233 444 664 532 221 1

as follows: 18 446 744 073 709 551 616 x                        8 --- 147 573 952 589 676 412 928 --- 63 355 33 525  74 414 14

Subtracting $1$ to get $M_{67}$:

Now to calculate $193 \, 707 \, 721 \times 761 \, 838 \, 257 \, 287$:

193 707 721         x 761 838 257 287 --             1 355 954 047             15 496 617 680             38 741 544 200          1 355 954 047 000          9 685 386 050 000         38 741 544 200 000      1 549 661 768 000 000      5 811 231 630 000 000    154 966 176 800 000 000    193 707 721 000 000 000 11 622 463 260 000 000 000 135 595 404 700 000 000 000 --- 147 573 952 589 676 412 927 ---  1 223 254 435 432 22  1

Thus:

as we were to demonstrate.