## Mersenne Prime Number Algorithm Error

### Yes, the 48th Mersenne Prime, 2^{57,885,161} – 1 is the largest known prime number.

Yet don't try and create it yourself, because the algorithm said to create it, is wrong!

From the awesome *Great Internet Mersenne Prime Search* project (GIMPS) media release we read, ** "The new prime number, 2 multiplied by itself 57,885,161 times, less one, has 17,425,170 digits".** Source: http://www.mersenne.org/primes/?press=M57885161

The statement the largest known prime number is *"2 multiplied by itself 57,885,161 times, less one"* is **incorrect**.

The correct algorithm is "2 multiplied by itself 57,885,16**0** times, less one"

The number "2 multiplied by itself 57,885,161 times, less one" is NOT a Mersenne Prime.

Here's why...

Just as two multiplied by itself 1 time is 2 × 2 or 2^{2}, the expression "2 multiplied by itself 57,885,161 times is, in fact, 2^{57885162}.

The exponent 57,885,162 is divisible by 2, so it is not prime and cannot be used to generate a Mersenne Prime, which takes the form (2^{n}) – 1 where n is prime. This was first documented in 1644 by Marin Mersenne, in the preface, section XIX, of * Minimi Cogitata Physico Mathematica*. (Click image to enlarge.)

Click here to read the relevant section.

Yet fast forward to today and 2^{57885162} – 1 is not a Mersenne Prime Number and *"2 multiplied by itself 57,885,161 times, less one"* is not a Mersenne Prime.

There are larger unknown primes than 2^{57885162} less 1. My laptop is hunting for a larger Mersenne Prime as I type this.

To become a Mesenne Prime Number hunter, goto http://www.mersenne.org/gettingstarted

Correct algorithms for the calculation of the largest known prime include:

*a*) "2 multiplied by itself 57,885,160 times, less one", or

*b*) "1 multiplied by two 57,885,161 times, less one".

The exponent in the formula **M***_{n}* =

**2**simply states how many times the unit (1) is doubled, before 1 is subtracted, to produce the prime. Grade 5 children can check the algorithm as follows.

*– 1*^{n}The first four Mersenne primes are : M_{2} = 3, M_{3} = 7, M_{5} = 31 and M_{7} = 127. We'll start with M_{2} = 3 which means 2^{2} – 1 = 3. This is mistakenly said to be "two multiplied by itself twice, minus one".

As children know, two multiplied by itself ONCE is 2 × 2, or 2^{2} or 4. Then 4 – 1 = 3.

OK, we will check why M_{3} = 7 because 2^{3}– 1 = 7.

Two multiplied by itself once = 2 × 2 = 2^{2} = 4, minus 1 = 3

Two multiplied by itself twice = 2 × 2 × 2 = 2^{3} = 8, minus 1 = 7 STOP!

We've hit 7 after two has been multiplied by itself TWICE, not three times.

In the absence of an initial multiplicative identity, 1, the correct algorithm for a Mersenne Prime in the form 2* ^{n}* – 1 is

Option a) 2 multiplied by itself *n* – 1 times, minus 1

Option b) 1 doubled *n* times minus 1.

CHECK M_{5}, or 2* ^{5}* – 1 = 31

One doubled one time, 1 × 2 = 2

^{1}= 2, minus 1 is 1.

One doubled 2 times, 1 × 2 × 2 = 2

^{2}= 4, minus 1 is 3. BINGO! M

_{2}is prime as 3 is prime

One doubled 3 times, 1 × 2 × 2 × 2 = 2

^{3}= 8, minus 1 is 7. BINGO! M

_{3}is prime as 7 is prime

One doubled four times, 1 × 2 × 2 × 2 × 2 = 2

^{4}= 16, minus 1 is 15. NOT PRIME

One doubled 5 times, 1 × 2 × 2 × 2 × 2 × 2 = 2

^{5}= 32, minus 1 is 31. BINGO! M

_{5}is prime as 31 is prime

**ERRATA IN THE DATA (CORRECTIONS BELOW)**

One doubled 57,885,161 times {1 x 2 x 2 x 2 x 2.... with 57,885,161 twos}, minus 1 is the largest prime yet discovered. OR.... 2 multiplied by itself 57,885,160 times, less one, is the largest prime yet discovered.

Elementary math teachers may like to get their students to check M_{7}, just as M_{5} has been checked above. This 'out-by-one' calculation error has been repeated on 13,000+ websites. That's bad news for anybody trying to understand simple arithmetic. (See link below.) https://www.google.ru/?gws_rd=ssl#newwindow=1&q=%222+multiplied+by+itself+57%2C885%2C161+times%2C+less+one%22

If you don't like the algorithm ** 1 doubled n times minus 1 **for a Mersenne Prime by

**M**

*=*

_{n}**2**, Gauss commenced his geometric progressions from 1, or unity. I may be the only person to say the commonly quoted algorithm for a Mersenne Prime Number is wrong, but if you disagree with me, you're disagreeing with Gauss! From

*– 1*^{n}*Disquisitiones Arithmeticae*, Gauss wrote the following (Article 45) in Latin,

*“In omni progressione geometrica 1, a, aa, a*In the case of Mersenne Primes,

^{3}, etc. praeter primum 1,…”*a*= 2.

**Source:**

*Disquisitiones Arithmeticae*, P. 38, Carl F. Gauss, 1801, translated as

*“In any geometric progression, 1, a, aa, a*on P. 47,

^{3}, etc. outside the first term 1, …”*Disquisitiones Arithmeticae,*by Arthur A. Clarke, Springer Verlag, New York, 1986.

The out-by-one error documented by the author was discussed in 1837, by Augustus De Morgan, who reprimanded children for the mistakes mathematicians make today! *"The beginner's common mistake is, that x multiplied n times by x is the nth power. This is not correct; x multiplied once by x (xx) is the second power; x multiplied n times by x is the (n + 1)th power." *

**Source:**

*Elements of Algebra Preliminary to the Differential Calculus*. A. De Morgan. P. 83, Taylor and Walton, London, 1837.

*Have fun, and remember, make you life count!*

Jonathan Crabtree

Mathematics Researcher

Melbourne Australia

www.jonathancrabtree.com/mathematics