Prime Decomposition of 9th Fermat Number

From ProofWiki
Jump to navigation Jump to search

Theorem

The prime decomposition of the $9$th Fermat number is given by:

\(\displaystyle 2^{\left({2^9}\right)} + 1\) \(=\) \(\displaystyle 13 \, 407 \, 807 \, 929 \, 942 \, 597 \, 099 \, 574 \, 024 \, 998 \, 205 \, 846 \, 127 \, 479 \, 365 \, 820 \, 592 \, 393 \, 377 \, 723 \, 561 \, 443 \, 721 \, 764 \, 030 \, 073 \, 546 \, 976 \, 801 \, 874 \, 298 \, 166 \, 903 \, 427 \, 690 \, 031 \, 858 \, 186 \, 486 \, 050 \, 853 \, 753 \, 882 \, 811 \, 946 \, 569 \, 946 \, 433 \, 649 \, 006 \, 084 \, 097\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \, 424 \, 833\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle 7 \, 455 \, 602 \, 825 \, 647 \, 884 \, 208 \, 337 \, 395 \, 736 \, 200 \, 454 \, 918 \, 783 \, 366 \, 342 \, 657\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle 741 \, 640 \, 062 \, 627 \, 530 \, 801 \, 524 \, 787 \, 141 \, 901 \, 937 \, 474 \, 059 \, 940 \, 781 \, 097 \, 519 \, 023 \, 905 \, 821 \, 316 \, 144 \, 415 \, 759 \, 504 \, 705 \, 008 \, 092 \, 818 \, 711 \, 693 \, 940 \, 737\)
\(\displaystyle \) \(=\) \(\displaystyle \left({1184 \times 2^{11} + 1}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({3 \, 640 \, 431 \, 067 \, 210 \, 880 \, 961 \, 102 \, 244 \, 011 \, 816 \, 628 \, 378 \, 312 \, 190 \, 597 \times 2^{11} + 1}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({362 \, 128 \, 936 \, 829 \, 849 \, 024 \, 182 \, 024 \, 971 \, 631 \, 805 \, 407 \, 255 \, 830 \, 459 \, 520 \, 272 \, 960 \, 891 \, 514 \, 314 \, 523 \, 640 \, 507 \, 570 \, 656 \, 742 \, 232 \, 821 \, 636 \, 569 \, 307 \times 2^{11} + 1}\right)\)


Proof


Also see


Historical Note

David Wells reports in his Curious and Interesting Numbers, 2nd ed. of $1997$ that this factorisation was accomplished by Arjen Klaas Lenstra and Mark Steven Manasse in $1990$.


Sources