Composite Mersenne Number/Examples/M227

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Example of Composite Mersenne Number

$M_{227}$ (that is, $2^{227} - 1$) is a composite number:

\(\displaystyle 2^{227} - 1\) \(=\) \(\displaystyle 215 \, 679 \, 573 \, 337 \, 205 \, 118 \, 357 \, 336 \, 120 \, 696 \, 157 \, 045 \, 389 \, 097 \, 155 \, 380 \, 324 \, 579 \, 848 \, 828 \, 881 \, 993 \, 727\)
\(\displaystyle \) \(=\) \(\displaystyle 26 \, 986 \, 333 \, 437 \, 777 \, 017 \times 7 \, 992 \, 177 \, 738 \, 205 \, 979 \, 626 \, 491 \, 506 \, 950 \, 867 \, 720 \, 953 \, 545 \, 660 \, 121 \, 688 \, 631\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 59 \, 441 \, 263 \, 078 \, 804 \times 227 + 1}\right) \times \left({2 \times 17 \, 603 \, 915 \, 722 \, 920 \, 659 \, 970 \, 245 \, 610 \, 023 \, 937 \, 711 \, 351 \, 422 \, 158 \, 858 \, 345 \times 227 + 1}\right)\)


Historical Note

Mersenne number $M_{227}$ was one of a set of $6$ demonstrated to be composite by Horace Scudder Uhler using a manual desk calculator in the $1940$s.