Composite Mersenne Number/Examples/M157

Example of Composite Mersenne Number

$M_{157}$ (that is, $2^{157} - 1$) is a composite number:

$\ds 2^{157} - 1$

$=$

$\ds 182 \, 687 \, 704 \, 666 \, 362 \, 864 \, 775 \, 460 \, 604 \, 089 \, 535 \, 377 \, 456 \, 991 \, 567 \, 871$

$\ds $

$=$

$\ds 852 \, 133 \, 201 \times 60 \, 726 \, 444 \, 167 \times 1 \, 654 \, 058 \, 017 \, 289 \times 2 \, 134 \, 387 \, 368 \, 610 \, 417$

$\ds $

$=$

$\ds \left({2 \times 2 \, 713 \, 800 \times 157 + 1}\right) \times \left({2 \times 193 \, 396 \, 319 \times 157 + 1}\right)$

$\ds $

$\, \ds \times \, $

$\ds \left({2 \times 5 \, 267 \, 700 \, 692 \times 157 + 1}\right) \times \left({2 \times 6 \, 797 \, 412 \, 001 \, 944 \times 157 + 1}\right)$

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 $M_{127}$.