2,147,483,647

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Number

$2 \, 147 \, 483 \, 647$ is:

A prime number

The $8$th Mersenne prime after $3$, $7$, $31$, $127$, $8191$, $131 \, 071$, $524 \, 287$:

$2 \, 147 \, 483 \, 647 = 2^{31} - 1$

The $11$th Mersenne number after $3$, $7$, $31$, $127$, $2047$, $8191$, $131 \, 071$, $524 \, 287$, $8 \, 388 \, 607$, $536 \, 870 \, 911$:

$2 \, 147 \, 483 \, 647 = 2^{31} - 1$

Historical Note

$2 \, 147 \, 483 \, 647$ was one of the Mersenne numbers that Marin Mersenne predicted to be prime in his Cogitata Physico-Mathematica of $1644$.

It was Leonhard Paul Euler who demonstrated it to be prime, which he did in $1772$.

Barlow's Prediction

Euler ascertained that $2^{31} - 1 = 2147483647$ is a prime number; and this is the greatest at present known to be such, and, consequently, the last of the above perfect numbers [that is, $2^{30}\left({2^{31} - 1}\right)$], which depends upon this, is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for, as they are merely curious without being useful, it is not likely that any person will attempt to find one beyond it.

This statement was made by Peter Barlow, in his $1811$ work Elementary Investigation of the Theory of Numbers.

He repeated this statement word for word in his $1814$ work A New Mathematical and Philosophical Dictionary.

Also see

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