proof that there are infinitely many primes using the Mersenne primes

proof that there are infinitely many primes using the Mersenne primes

Like Euclid’s classic proof, this is also a

proof by contradiction

which starts out from the

assumption

that there is a largest

prime number

. But instead of building a

primorial

or a

factorial

, a

Mersenne number

is computed instead.

Proof.

Assume that the

odd prime

p

is the largest prime number. Then

2

p

-

1

is

composite

, not a

Mersenne prime

. So what is the

integer factorization

of

2

p

-

1

? By Fermat’s little

theorem

we have

2

p

-

1

≡

1

mod

p

, doubling that we get

2

p

≡

2

mod

p

, therefore

2

p

-

1

mod

p

. Since

2

p

-

1

is composite, it must have at least two

prime factors

.

If a prime number

q

is a factor of

2

p

-

1

(that is,

2

p

-

1

≡

0

mod

q

) then, since

2

q

-

1

-

1

≡

0

mod

q

, assigning

d

=

gcd

⁡

(

p

,

q

-

1

)

, we have

2

d

-

1

≡

0

mod

q

. Then

d

can not be 1 (for 2 - 1 is not

congruent

to 0 modulo

q

). Therefore

d

=

p

and

q

-

1

is

multiple

of

p

, and thus we derive that

q

=

n

⁢

p

+

1

.

Because we stipulated that

p

is an odd prime and

2

p

-

1

is also an

odd number

,

n

in the

formula

n

⁢

p

+

1

must be an

even number

, so we can then set

m

=

n

2

, giving us the form of the prime factors of

2

p

-

1

as

2

⁢

m

⁢

p

+

1

.

So now we have to solve for

m

. There must be solutions in the range

0

<

m

<

(

2

p

-

1

-

1

)

. If

m

=

1

, that gives us

2

⁢

p

+

1

as a prime factor of

2

p

-

1

, but since

(

2

⁢

p

+

1

)

>

p

(regardless of which odd prime

p

is), that also contradicts our first statement that

p

is the largest prime. With any larger

m

, the number

2

⁢

m

⁢

p

+

1

is even larger than

p

, and thus our initial statement is contradicted regardless of the

primality

of

2

p

-

1

.
∎

Two examples to show both scenarios: first,

p

=

29

is the largest prime. The corresponding Mersenne number is 536870911, which must be composite. In fact, its factorization, stated in

relation

to 2 and 29, is

(

1

+

2

×

4

×

29

)

⁢

(

1

+

2

×

19

×

29

)

⁢

(

1

+

2

×

36

×

29

)

. Already the least prime factor, 233, is larger than 29, thus 29 can’t be the largest prime.

Second example,

p

=

31

is the largest prime. The corresponding Mersenne number is 2147483647, which must be composite. But a search for solutions to

(

2

m

p

+

1

)

|

(

2

p

-

1

)

, solving for

m

, yields only two solutions,

m

=

0

and

m

=

2

p

-

1

-

1

, corresponding to 1 and

2

p

-

1

. This means 2147483647 is

divisible

only by 1 and itself, and is therefore prime, and is in fact a much larger prime than 31.

Note, however, that while this proves there are infinitely many primes, it does not resolve the issue of whether or not there are infinitely many Mersenne primes.

Title

proof that there are infinitely many primes using the Mersenne primes

Canonical name

ProofThatThereAreInfinitelyManyPrimesUsingTheMersennePrimes

Date of creation

2013-03-22 18:05:14

Last modified on

2013-03-22 18:05:14

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Proof

Classification

msc 11A41

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