Prime Decomposition of 9th Fermat Number
Theorem
The prime decomposition of the $9$th Fermat number is given by:
| \(\ds 2^{\paren {2^9} } + 1\) | \(=\) | \(\ds 13 \, 407 \, 807 \, 929 \, 942 \, 597 \, 099 \, 574 \, 024 \, 998 \, 205 \, 846 \, 127 \, 479 \, 365 \, 820 \, 592 \, 393 \, 377 \, 723 \, 561 \, 443 \, 721 \, 764 \, 030 \, 073 \, 546 \, 976 \, 801 \, 874 \, 298 \, 166 \, 903 \, 427 \, 690 \, 031 \, 858 \, 186 \, 486 \, 050 \, 853 \, 753 \, 882 \, 811 \, 946 \, 569 \, 946 \, 433 \, 649 \, 006 \, 084 \, 097\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \, 424 \, 833\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds 7 \, 455 \, 602 \, 825 \, 647 \, 884 \, 208 \, 337 \, 395 \, 736 \, 200 \, 454 \, 918 \, 783 \, 366 \, 342 \, 657\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds 741 \, 640 \, 062 \, 627 \, 530 \, 801 \, 524 \, 787 \, 141 \, 901 \, 937 \, 474 \, 059 \, 940 \, 781 \, 097 \, 519 \, 023 \, 905 \, 821 \, 316 \, 144 \, 415 \, 759 \, 504 \, 705 \, 008 \, 092 \, 818 \, 711 \, 693 \, 940 \, 737\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2^5 \times 37 \times 2^{11} + 1}\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {19 \times 47 \times 82 \, 488 \, 781 \times 1 \, 143 \, 290 \, 228 \, 161 \, 321 \times 43 \, 226 \, 490 \, 359 \, 557 \, 706 \, 629 \times 2^{11} + 1}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {1129 \times 26 \, 813 \times 40 \, 644 \, 377 \times 17 \, 338 \, 437 \, 577 \, 121 \times 16 \, 975 \, 143 \, 302 \, 271 \, 505 \, 426 \, 897 \, 585 \, 653 \, 131 \, 126 \, 520 \, 182 \, 328 \, 037 \, 821 \, 729 \, 720 \, 833 \, 840 \, 187 \, 223 \times 2^{11} + 1}\) |
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
- Jul. 1993: A.K. Lenstra, H.W. Lenstra, Jr., M.S. Manasse and J.M. Pollard: The Factorization of the Ninth Fermat Number (Math. Comp. Vol. 61, no. 203: pp. 319 – 349) www.jstor.org/stable/2152957
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2^{2^9} + 1 = 2^{512} + 1$