Prime Decomposition of 10th Fermat Number
| \(\ds 2^{\paren {2^{10} } } + 1\) | \(=\) | \(\ds 179 \, 769 \, 313 \, 486 \, 231 \, 590 \, 772 \, 930 \, 519 \, 078 \, 902 \, 473 \, 361 \, 797 \, 697 \, 894 \, 230 \, 657 \, 273 \, 430 \, 081 \, 157 \, 732 \, 675 \, 805 \, 500 \, 963 \, 132 \, 708 \, 477 \, 322 \, 407 \, 536 \, 021 \, 120 \, 113 \, 879 \, 871 \ldots\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \ldots \, \) | \(\ds 393 \, 357 \, 658 \, 789 \, 768 \, 814 \, 416 \, 622 \, 492 \, 847 \, 430 \, 639 \, 474 \, 124 \, 377 \, 767 \, 893 \, 424 \, 865 \, 485 \, 276 \, 302 \, 219 \, 601 \, 246 \, 094 \, 119 \, 453 \, 082 \, 952 \, 085 \, 005 \, 768 \, 838 \, 150 \, 682 \, 342 \, 462 \ldots\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \ldots \, \) | \(\ds 881 \, 473 \, 913 \, 110 \, 540 \, 827 \, 237 \, 163 \, 350 \, 510 \, 684 \, 586 \, 298 \, 239 \, 947 \, 245 \, 938 \, 479 \, 716 \, 304 \, 835 \, 356 \, 329 \, 624 \, 224 \, 137 \, 217\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 45 \, 592 \, 577\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds 6 \, 487 \, 031 \, 809\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds 4 \, 659 \, 775 \, 785 \, 220 \, 018 \, 543 \, 264 \, 560 \, 743 \, 076 \, 778 \, 192 \, 897\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds 130 \, 439 \, 874 \, 405 \, 488 \, 189 \, 727 \, 484 \, 768 \, 796 \, 509 \, 903 \, 946 \, 608 \, 530 \, 841 \, 611 \, 892 \, 186 \, 895 \, 295 \, 776 \, 832 \, 416 \, 251 \, 471 \, 863 \, 574 \, 140 \, 227 \, 977 \, 573 \, 104 \, 895 \, 898 \, 783 \, 928 \, 842 \ldots\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \ldots \, \) | \(\ds 923 \, 844 \, 831 \, 149 \, 032 \, 913 \, 798 \, 729 \, 088 \, 601 \, 617 \, 946 \, 094 \, 119 \, 449 \, 010 \, 595 \, 906 \, 710 \, 130 \, 531 \, 906 \, 171 \, 018 \, 354 \, 491 \, 609 \, 619 \, 193 \, 912 \, 488 \, 538 \, 116 \, 080 \, 712 \, 299 \, 672 \, 322 \ldots\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \ldots \, \) | \(\ds 806 \, 217 \, 820 \, 753 \, 127 \, 014 \, 424 \, 577\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {11 \, 131 \times 2^{12} + 1}\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {2^2 \times 3^2 \times 29 \times 37 \times 41 \times 2^{12} + 1}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {3 \times 5 \, 639 \times 8 \, 231 \times 433 \, 639 \times 18 \, 840 \, 862 \, 799 \, 165 \, 386 \, 003 \, 967 \times 2^{12} + 1}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {2 \times 3 \times 13 \times 23 \times 29 \times 6 \, 329 \times 760 \, 347 \, 109 \times N_{231} \times 2^{12} + 1}\) |
Also see
Historical Note
$F_{10}$ was proved composite in $1952$ by Raphael Mitchel Robinson using Pépin's Test on the SWAC, but at that time the factors had yet to be determined.
John Lewis Selfridge discovered the factor $45 \, 592 \, 577$ in $1953$, also using the SWAC.
At the same time he discovered the factor $825 \, 753 \, 601$ of $F_{16}$.
John David Brillhart discovered the factor $6 \, 487 \, 031 \, 809$ in $1962$ on an IBM 704.
Brillhart later found that the cofactor was a $291$-digit composite number.
The factors of this $291$-digit composite were finally found by Richard Peirce Brent in $1995$.
As he explained in his $1999$ article, the reason why the factors of $F_{11}$ were found so much earlier is that the second largest factor of $F_{11}$ had a mere $22$ digits, as opposed to the $40$ digits of that of $F_{10}$.
It may be noted that the author of this page performed this exercise in $2020$ using a factorisation tool freely available online, running on a machine of modest specifications.
It took just under $7$ hours in total.
Sources
- Jan. 1999: Richard P. Brent: Factorization of the Tenth Fermat Number (Math. Comp. Vol. 68, no. 235: pp. 429 – 451) www.jstor.org/stable/2585124