Download Number Theory III: Mersenne and Fermat Type Numbers

Math 347, Fall 2016
Number Theory III: Mersenne and Fermat Type Numbers
A.J. Hildebrand
Number Theory III: Mersenne
and Fermat Type Numbers
Among the most famous types of numbers are the Fermat and Mersenne numbers, defined as follows:
• A Mersenne number is a number of the form Mn = 2n − 1, where n ∈ N.
k
• A Fermat number is a number of the form Fk = 22 + 1, where k ∈ N.
Mersenne and Fermat primes
The main reason for the interest in these numbers is in the fact that numbers of this type appear to be good
candidates for primes:
• Mersenne primes: The first four Mersenne numbers with prime index n turn out to be all primes:
M2 = 22 − 1 = 3, M3 = 23 − 1 = 7, M5 = 25 − 1 = 31, M7 = 27 − 1 = 127. This might lead one to
conjecture that all Mersenne numbers in which the index n is prime, are prime. However, M11 turns
out to be composite, so this conjecture is not true.
• Record Mersenne primes: Nearly all of the record primes discovered over the past 50 years have
been of the form of a Mersenne prime: Famous examples include:
• The “postmark prime”, M11213 = 211213 − 1: This prime was discovered in 1963 at the U of
I by Professor Don Gillies with the ILLIAC II computer. It was the largest known prime at the
time. The discovery was considered sufficiently significant that a special postmark (shown below)
was created to celebrate the event.
• The $100,000 prime”, M43,112,609 = 2431126009 − 1: This was the first prime number found with
more than 10, 000, 000 digits. The research group that discovered it earned an award of $100,000.
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• Fermat primes: The first four Fermat numbers are all primes: F1 = 22 − 1 = 5, F2 = 22 − 1 = 17,
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F3 = 22 − 1 = 257, F4 = 22 − 1 = 65537. However, the next one, F5 = 4, 294, 927, 297, turns out to
be composite; in fact, every single Fermat number beyond F4 that has been tested has been found to
be composite.
*** See back of page for key results and proofs about numbers 2n ± 1 ***
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Math 347, Fall 2016
Number Theory III: Mersenne and Fermat Type Numbers
A.J. Hildebrand
Primality/compositeness of numbers of the form 2n − 1
The reason why one is interested in numbers of the form 2n − 1 with prime n, is that those are the only
cases that can possibly yield a prime. We have:
Theorem (Primality/compositeness of Mersenne type numbers). If n is composite, then 2n − 1 is composite as well. Hence, the only possible primes of the form
2n − 1 are those where the exponent n is a prime number.
Proof: We break the proof into several steps:
• Claim 1. Let a, n ∈ N with a ≥ 2. Then an − 1 is divisible by a − 1.
Proof. We have a ≡ 1 mod a − 1, hence an ≡ 1n = 1 mod a − 1, and therefore a − 1|an − 1.
• Claim 2: Let a, n ∈ N with a ≥ 3 and n ≥ 2. Then an − 1 is composite. Thus, the only numbers of the form an − 1 with exponent n ≥ 2 that are possibly prime are those of the form 2n − 1.
Proof: By Claim 1, an − 1 is divisible by a − 1. The given conditions, a ≥ 3 and n ≥ 2, imply that
a − 1 ≥ 2 and a − 1 < an − 1, so a − 1 is a nontrivial divisor of an − 1. Therefore an − 1 is composite.
• Claim 3: If n is composite, then 2n − 1 is also composite. Thus, the only numbers of the form
2n − 1 that are possibly prime are those where the exponent itself is prime.
Proof: Suppose n is composite. Then there exist h, k ∈ N with 1 < h, k < n such that n = hk.
Therefore 2n − 1 = 2hk − 1 = (2h )k − 1. Applying the result of Claim 2 with a = 2h and n = k (since
h ≥ 2, the condition “a ≥ 3” holds, and since k ≥ 2, the condition “n ≥ 2” holds), we conclude that
2n − 1 is composite.
Primality/compositeness of numbers of the form 2n + 1
An analogous result holds for numbers of the form 2n + 1. The only cases that can possibly yield primes are
those when n is a power of 2.
Theorem (Primality/compositeness of Fermat type numbers). If n is not a
power of 2, then 2n + 1 is composite. Hence, the only possible primes of the form 2n + 1
k
are those where the exponent n is a power of 2, i.e., the Fermat numbers Fk = 22 + 1.
Proof: We break the proof into several steps:
• Claim 1:
Let a, n ∈ N with n odd. Then an + 1 is divisble by a + 1.
Proof. We have a ≡ −1 mod a+1, hence an ≡ (−1)n = −1 mod a−1 (since n is odd), and therefore
a + 1|an + 1.
• Claim 2: Let a, n ∈ N with n odd, a ≥ 2 and n ≥ 3. Then an + 1 is composite.
Proof: By Claim 1, an + 1 is divisible by a + 1. By the assumptions in Claim 2, a ≥ 2 and n ≥ 3,
we have a + 1 ≥ 3 and a + 1 < an + 1, so a + 1 is a nontrivial divisor of an + 1. Therefore an + 1 is
composite.
• Claim 3: If n is an integer ≥ 2 that is not a power of 2, then 2n + 1 is composite. Thus,
the only numbers of the form 2n + 1 that are possibly prime are those where the exponent n is a power
k
of 2, i.e., the Fermat numbers Fk = 22 + 1.
Proof: Suppose n is an integer ≥ 2 that is not a power of 2. Then n can be written as n = 2k m, where
k
k is a nonnegative integer and m an odd integer with m ≥ 3. Then 2n + 1 = (22 )m + 1. Applying the
k
result of Claim 2 with a = 22 and n = m then shows that 2n + 1 = am + 1 is composite.
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