Composite Mersenne Number/Examples/M199

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Example of Composite Mersenne Number

$M_{199}$ (that is, $2^{199} - 1$) is a composite number:

\(\displaystyle 2^{199} - 1\) \(=\) \(\displaystyle 803 \, 469 \, 022 \, 129 \, 495 \, 137 \, 770 \, 981 \, 046 \, 170 \, 581 \, 301 \, 261 \, 101 \, 496 \, 891 \, 396 \, 417 \, 650 \, 687\)
\(\displaystyle \) \(=\) \(\displaystyle 164 \, 504 \, 919 \, 713 \times 4 \, 884 \, 164 \, 093 \, 883 \, 941 \, 177 \, 660 \, 049 \, 098 \, 586 \, 324 \, 302 \, 977 \, 543 \, 600 \, 799\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 413 \, 328 \, 944 \times 199 + 1}\right) \times \left({2 \times 12 \, 271 \, 769 \, 080 \, 110 \, 404 \, 968 \, 995 \, 098 \, 237 \, 654 \, 081 \, 163 \, 260 \, 159 \, 801 \times 199 + 1}\right)\)


Historical Note

Mersenne number $M_{199}$ was one of a set of $6$ demonstrated to be composite by Horace Scudder Uhler using a manual desk calculator in the $1940$s, in what turned out to be a vain attempt to find the next Mersenne prime after $M_{127}$.