# Composite Mersenne Number/Examples/M257

## Example of Composite Mersenne Number

$M_{257}$ (that is, $2^{257} - 1$) is a composite number:

 $\ds 2^{257} - 1$ $=$ $\ds 231 \, 584 \, 178 \, 474 \, 632 \, 390 \, 847 \, 141 \, 970 \, 017 \, 375 \, 815 \, 706 \, 539 \, 969 \, 331 \, 281 \, 128 \, 078 \, 915 \, 168 \, 015 \, 826 \, 259 \, 279 \, 871$ $\ds$ $=$ $\ds 535 \, 006 \, 138 \, 814 \, 359 \times 1 \, 155 \, 685 \, 395 \, 246 \, 619 \, 182 \, 673 \, 033 \times 374 \, 550 \, 598 \, 501 \, 810 \, 936 \, 581 \, 776 \, 630 \, 096 \, 313 \, 181 \, 393$ $\ds$ $=$ $\ds \paren {2 \times 1 \, 040 \, 867 \, 974 \, 347 \times 257 + 1}$ $\ds$  $\, \ds \times \,$ $\ds \paren {2 \times 2 \, 248 \, 415 \, 165 \, 849 \, 453 \, 662 \, 788 \times 257 + 1}$ $\ds$  $\, \ds \times \,$ $\ds \paren {2 \times 728 \, 697 \, 662 \, 454 \, 885 \, 090 \, 626 \, 024 \, 572 \, 171 \, 815 \, 528 \times 257 + 1}$

## Historical Note

Mersenne number $M_{257}$ was the largest Mersenne number conjectured by Marin Mersenne to be prime, in his Cogitata Physico-Mathematica of $1644$.

In $1922$, Maurice Kraitchik demonstrated that it was composite, but did not find any actual factors.

Its full factorisation was eventually achieved by M.A. Penk and Robert Baillie.

It is instructive to note that, in $2017$, it took a freely-available online factorization tool $14 \cdotp 8$ seconds to perform this exact calculation.