Mersenne was interested in numbers of the form 2^N-1 (that is multiply 2 together N times and subtract 1). He thought they might often be prime numbers. Mersenne was fascinated by music. It is possible that he took his inspiration for looking at these numbers from what happens to a note when you double the frequency. Doubling a note makes a note an octave higher – a harmonious note. Take one away and suddenly the note becomes discordant – a prime note maybe?

When is 2^N-1 a prime number? Here is a list for the first N between 1 and 20.

N 2N-1 Prime? Check 1 1 Yes   No 2 3 Yes   No 3 7 Yes   No 4 15 Yes   No 5 31 Yes   No 6 63 Yes   No 7 127 Yes   No 8 255 Yes   No 9 511 Yes   No 10 1023 Yes   No 11 2047 Yes   No 12 4095 Yes   No 13 8191 Yes   No 14 16383 Yes   No 15 32767 Yes   No 16 65535 Yes   No 17 131071 Yes   No 18 262143 Yes   No 19 524287 Yes   No 20 1047575 Yes   No

	Click here to find out why 1 is not a prime.

If N is not prime then 2^N-1 cannot be prime. Why?

Click here to find out why if N=pxq then there is a way to factorize 2N-1.

But unfortunately even if N is prime then that doesn’t guarantee that 2^N-1 will be prime. For example
2¹¹-1=2047=23x89.

Most of the record breaking biggest primes that have been discovered are Mersenne primes.

Click here to find out which N have been discovered which produce primes…- this takes you to 1.8

Click here to find out which N have been discovered which produce primes…