Composite Mersenne Number/Examples/M167

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Example of Composite Mersenne Number

$M_{167}$ (that is, $2^{167} - 1$) is a composite number:

\(\displaystyle 2^{167} - 1\) \(=\) \(\displaystyle 187 \, 072 \, 209 \, 578 \, 355 \, 573 \, 530 \, 071 \, 658 \, 587 \, 684 \, 226 \, 515 \, 959 \, 365 \, 500 \, 927\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \, 349 \, 023 \times 79 \, 638 \, 304 \, 766 \, 856 \, 507 \, 377 \, 778 \, 616 \, 296 \, 087 \, 448 \, 490 \, 695 \, 649\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 7033 \times 167 + 1}\right) \times \left({2 \times 238 \, 438 \, 038 \, 224 \, 121 \, 279 \, 574 \, 187 \, 473 \, 940 \, 381 \, 582 \, 307 \, 472 \times 167 + 1}\right)\)


Historical Note

Mersenne number $M_{167}$ 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 $M_{127}$.