MATH 035
Penn State University
Dr. James Sellers
Handout: Twin Prime Conjecture / Prime Number Theorem
In our previous handout, we saw that we can find arbitrarily large gaps between consecutive
primes (or, said another way, we can find arbitrarily long lists of consecutive non-prime
numbers). We begin today’s handout by looking at a somewhat “opposite” idea: how close can
we make primes, and how often can we do so? We start with a definition.
Definition: Let p be a prime number such that p+2 is also prime. Then we say that p and p+2
are twin primes.
Examples: 17 and 19 are twin primes. 101 and 103 are twin primes.
Notice that, except in the case of the numbers 2 and 3, you cannot get primes any closer than
two apart from one another (like 17 and 19). That’s because the number between them (in this
case 18) is even and therefore divisible by 2. So twin primes really do represent primes that are
as close to one another as possible.
So this begs a question: How many pairs of twin primes are there? The answer is that we do
not know at this time! However, most mathematicians would agree with the following
conjecture:
Twin Prime Conjecture: There are infinitely many pairs of twin primes.
Remember, a conjecture is simply a guess (usually backed by some sort of empirical evidence).
So we don’t have a proof of this statement! But there is lots of evidence in our favor. For
example, there are 35 pairs of twin primes below 1000; the first is (3,5) and the last is
(881,883). There are 808,675,888,577,436 twin prime pairs below 1018. The largest known pair
of twin primes, found in 2009, is 65516468355 · 2333333 ± 1. The numbers have 100355 decimal
digits.
There is one other result on the distribution of primes which we should consider today. It is, in
some sense, the best possible type of result we can expect regarding the primes. Because of
its importance, it has a special name: The Prime Number Theorem.
Theorem (The Prime Number Theorem): Let x be a positive integer (which is usually thought
of as a very LARGE positive integer). Then the number of prime numbers less than or equal to
x
x is asymptotically the same as the value
.
ln x
So let’s make sure we understand what this says and what it does NOT say. First, it does NOT
x
say that the number of primes less than or equal to x EQUALS
. What it does say is that,
ln x
as x gets larger and larger, the number of primes less than or equal to x gets closer and closer
x
.
to the quantity
ln x
Let’s look at some empirical data that backs up this claim. A nice table demonstrating this can
be found at http://en.wikipedia.org/wiki/Prime_Number_Theorem .
© 2010, James A. Sellers