Composite Mersenne Number/Examples/M193

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Example of Composite Mersenne Number

$M_{193}$ (that is, $2^{193} - 1$) is a composite number:

\(\displaystyle 2^{193} - 1\) \(=\) \(\displaystyle 12 \, 554 \, 203 \, 470 \, 773 \, 361 \, 527 \, 671 \, 578 \, 846 \, 415 \, 332 \, 832 \, 204 \, 710 \, 888 \, 928 \, 069 \, 025 \, 791\)
\(\displaystyle \) \(=\) \(\displaystyle 13 \, 821 \, 503 \times 61 \, 654 \, 440 \, 233 \, 248 \, 340 \, 616 \, 559 \times 14 \, 732 \, 265 \, 321 \, 145 \, 317 \, 331 \, 353 \, 282 \, 383\)
\(\displaystyle \) \(=\) \(\displaystyle \left({2 \times 35 \, 807 \times 193 + 1}\right) \times \left({2 \times 159 \, 726 \, 529 \, 101 \, 679 \, 638 \, 903 \times 193 + 1}\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle \left({2 \times 38 \, 166 \, 490 \, 469 \, 288 \, 386 \, 868 \, 790 \, 887 \times 193 + 1}\right)\)


Historical Note

Mersenne number $M_{193}$ 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 $M_{127}$.