Composite Mersenne Number/Examples/M229
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Example of Composite Mersenne Number
$M_{229}$ (that is, $2^{229} - 1$) is a composite number:
| \(\ds 2^{229} - 1\) | \(=\) | \(\ds 862 \, 718 \, 293 \, 348 \, 820 \, 473 \, 429 \, 344 \, 482 \, 784 \, 628 \, 181 \, 556 \, 388 \, 621 \, 521 \, 298 \, 319 \, 395 \, 315 \, 527 \, 974 \, 911\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \, 504 \, 073 \times 20 \, 492 \, 753 \times 59 \, 833 \, 457 \, 464 \, 970 \, 183 \times 467 \, 795 \, 120 \, 187 \, 583 \, 723 \, 534 \, 280 \, 000 \, 348 \, 743 \, 236 \, 593\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \times 3284 \times 229 + 1}\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {2 \times 44744 \times 229 + 1}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {2 \times 130 \, 640 \, 736 \, 823 \, 079 \times 229 + 1}\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \times \, \) | \(\ds \paren {2 \times 1 \, 021 \, 386 \, 725 \, 300 \, 401 \, 143 \, 087 \, 947 \, 599 \, 014 \, 723 \, 224 \times 229 + 1}\) |
Historical Note
Mersenne number $M_{229}$ was the largest of a set of $6$ demonstrated to be composite by Horace Scudder Uhler using a manual desk calculator in the $1940$s.
The next Mersenne prime after $M_{127}$, proved prime in $1896$ by François Édouard Anatole Lucas, does not occur until $M_{521}$.
As Uhler remarked, nobody had any idea that the next one would be so far away.
It is instructive to note that, in $2017$, it took a freely-available online factorization tool $4 \cdotp 8$ seconds to perform this exact calculation.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2^{229} - 1$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2^{229} - 1$