Prime Decomposition of 9th Fermat Number

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Theorem

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

\(\displaystyle 2^{\left({2^9}\right)} + 1\) \(=\) \(\displaystyle 13 \, 407 \, 807 \, 929 \, 942 \, 597 \, 099 \, 574 \, 024 \, 998 \, 205 \, 846 \, 127 \, 479 \, 365 \, 820 \, 592 \, 393 \, 377 \, 723 \, 561 \, 443 \, 721 \, 764 \, 030 \, 073 \, 546 \, 976 \, 801 \, 874 \, 298 \, 166 \, 903 \, 427 \, 690 \, 031 \, 858 \, 186 \, 486 \, 050 \, 853 \, 753 \, 882 \, 811 \, 946 \, 569 \, 946 \, 433 \, 649 \, 006 \, 084 \, 097\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \, 424 \, 833\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle 7 \, 455 \, 602 \, 825 \, 647 \, 884 \, 208 \, 337 \, 395 \, 736 \, 200 \, 454 \, 918 \, 783 \, 366 \, 342 \, 657\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle 741 \, 640 \, 062 \, 627 \, 530 \, 801 \, 524 \, 787 \, 141 \, 901 \, 937 \, 474 \, 059 \, 940 \, 781 \, 097 \, 519 \, 023 \, 905 \, 821 \, 316 \, 144 \, 415 \, 759 \, 504 \, 705 \, 008 \, 092 \, 818 \, 711 \, 693 \, 940 \, 737\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {2^5 \times 37 \times 2^{11} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \paren {19 \times 47 \times 82 \, 488 \, 781 \times 1 \, 143 \, 290 \, 228 \, 161 \, 321 \times 43 \, 226 \, 490 \, 359 \, 557 \, 706 \, 629 \times 2^{11} + 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \paren {1129 \times 26 \, 813 \times 40 \, 644 \, 377 \times 17 \, 338 \, 437 \, 577 \, 121 \times 16 \, 975 \, 143 \, 302 \, 271 \, 505 \, 426 \, 897 \, 585 \, 653 \, 131 \, 126 \, 520 \, 182 \, 328 \, 037 \, 821 \, 729 \, 720 \, 833 \, 840 \, 187 \, 223 \times 2^{11} + 1}\)


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