21
Jan 16

## The world's largest known prime number, called a Mersenne Prime, is 1 repeated 74,207,280 times, in base 2.

So memorise 74,207,280 and you can claim to have memorised the world's largest known prime; in base 2.

In Base 2, it's 11111111111111... with 74,207,280 ones in a row!

BTW, it's NOT, "two multiplied by itself 74,207,281 times, minus 1" as widely reported online.

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Until last week, the world's largest known prime number was 2⁵⁷ ⁸⁸⁵ ¹⁶¹ – 1. The media release that announced the discovery, stated the algorithm behind this prime was, "2 multiplied by itself 57,885,161 times, less one".

It wasn't, although it seems to have been endorsed by mathematics professors.

The number two, multiplied by itself 1 time is 2 × 2 or 2².
The number two, multiplied by itself 3 times  is 2 × 2 × 2 × 2 or 2⁴.
The number two, multiplied by itself 5 times is 2 × 2 × 2 × 2 × 2 × 2 or 2⁶.

See the pattern?

When the number of times a base is multiplied by itself, is odd, the exponent is even! So 2 multiplied by itself 57,885,161 times is  actually 2⁵⁷ ⁸⁸⁵ ¹⁶².

Because the exponent ⁵⁷ ⁸⁸⁵ ¹⁶² is even, 2 multiplied by itself 57,885,161 times – 1 cannot be a Mersenne Prime!

A Mersenne Prime takes the form 2ᴾ – 1  where P is a prime number.

The exponent you get when you multiply 2 by itself 57885161 times is 57885162 and this can be factored by 2, 3, 6, 9647527, 19295054 and 28942581.  So "2 multiplied by itself 57,885,161 times, less one" is not a Mersenne Prime, despite the official media release giving that "multiplied by itself" explanation!

The scientific method has got to be admired though. You see I wrote to Dr Curtis Cooper to let him know the correct explanations was either...

"2 multiplied by itself 57,885,16o times, less one"

or

"the product of 57,885,161 twos multiplied together, less one."

So I was very pleased to read the media release this week. It gave a correct explanation of the world's new largest known prime number!

"calculated by multiplying together 74,207,281 twos then subtracting one."

A simpler explanation for kids is...

"double one 74,207,281 times, in succession, then subtract one!"

SOURCES:

http://www.mersenne.org/primes/?press=M74207281