9
Jun 15

Mersenne Prime Number Algorithm Error

Yes, the 48th Mersenne Prime, 257,885,161 – 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 57,885,160 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 22, the expression "2 multiplied by itself 57,885,161 times is, in fact, 257885162.

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 (2n) – 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.)

1644-MMERSENNE_Minimi_Cogitata_Physico_Mathematica_Title_Page

Click here to read the relevant section.

Yet fast forward to today and 257885162 – 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 257885162 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 Mn = 2n  1 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.

The first four Mersenne primes are : M2 = 3, M3 = 7, M5 = 31 and M7 = 127. We'll start with M2 = 3 which means 22  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 22 or 4. Then 4  1 = 3.

OK, we will check why M3 = 7 because 23 1 = 7.
Two multiplied by itself once = 2 × 2 = 22 = 4, minus 1 = 3
Two multiplied by itself twice = 2 × 2 × 2 = 23 = 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 2n  1 is

Option a) 2 multiplied by itself n 1 times, minus 1
Option b) 1 doubled n times minus 1.

CHECK M5, or 25  1 = 31
One doubled one time, 1 × 2 = 21 = 2, minus 1 is 1.
One doubled 2 times, 1 × 2 × 2 = 22 = 4, minus 1 is 3. BINGO!  M2 is prime as 3 is prime
One doubled 3 times, 1 × 2 × 2 × 2 = 23 = 8, minus 1 is 7. BINGO! M3 is prime as 7 is prime
One doubled four times, 1 × 2 × 2 × 2 × 2 = 24 = 16, minus 1 is 15. NOT PRIME
One doubled 5 times, 1 × 2 × 2 × 2 × 2 × 2 = 25 = 32, minus 1 is 31.  BINGO! M5 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 M7, just as M5 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 Mn = 2n  1, 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 Disquisitiones Arithmeticae, Gauss wrote the following (Article 45) in Latin, “In omni progressione geometrica 1, a, aa, a3, etc. praeter primum 1,…”  In the case of Mersenne Primes, a = 2. Source: Disquisitiones Arithmeticae, P. 38, Carl F. Gauss, 1801, translated as “In any geometric progression, 1, a, aa, a3, etc. outside the first term 1, …” on P. 47, 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