Skip to main content

Prime numbers - Do they end at a point?

Prime numbers are one of the most speculated topics in mathematics. Mathematicians have pondered about them enough to arrive at a proper conclusion to the above question.

Let us look at the first few prime numbers:
$$2\quad3\quad5\quad7\quad11\quad13\quad17\quad19\quad23\quad29\quad31\quad37\quad41$$ Let us now look at the first few prime numbers after $100000$:
$$100003\quad100019\quad100043\quad100049\quad100057\quad100069\quad100103\\100109\quad100129\quad100151\quad100153\quad100169\quad100183$$ If you look closely, you may notice that the average separation between consecutive prime numbers between $100003$ and $100183$ is larger than that of $2$ and $41$. Naturally, our brain thinks that this separation increases on and on as we go to larger numbers, and at one point the set of prime numbers terminate. But that is not the case. It is proven that there are infinitely many prime numbers.

The Proof:

Euclid had the same question we have today, and he gave a beautiful proof that is quoted even today for proving that the set of prime numbers do not terminate.

Let us assume that there are only a finite number of primes, name that set $P$.
$$P = \lbrace a_1, a_2, a_3,...,a_n \rbrace$$
Consider a number $A$ such that
$$A = a_1\times a_2\times a_3\times a_4\times ...\times a_n + 1$$
Since the prime number list ended at $a_n$, $A$ must not be a prime number. So $A$ must have a factor that belongs to the set $P$.

Let $a_m$ be the factor of $A$ where $1\le m\le n$. But it is clear that when $A$ is divided by $a_m$, it leaves a remainder 1! But this contradicts the fact that factors leave a remainder of 0 when dividing the number.

Thus, the original assumption that there is a finite list of prime numbers is false. Hence it is proved by the method of contradiction that there are infinitely many prime numbers.
Labels:

Comments

Post a Comment

Popular posts from this blog

The $\operatorname{erf}$ function

Can a function be 'defined' as the anti-derivative of another function? $\operatorname{erf}$ is one such function. Even though it has various proper infinite series expansions (as in Wolfram MathWorld - Erf ), it is defined by mathematicians as such: $$\operatorname{erf}(z) \equiv \frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}dt$$ This function has extensive implications in statistics, and can be used to express the integral of $e^{-x^2}$ and $e^{x^2}$. See the post that tries to find  integral of $e^{x^2}$  - we need not use integration by parts there, but rather can use the $\operatorname{erf}$ function. Source: Wolfram MathWorld
Post a Comment
Read more

Factorials, and the Gamma Function

What is the Gamma function? What does it have to do with the factorial of a number? And why does such a formula even work? The Factorial In our high school combinatorics course, we were introduced to the factorial of a natural number. We defined the factorial of a number $n$ to be the product of all the natural numbers till $n$. Or symbolically: $$n! = 1 . 2 . 3 ... (n-2)  (n-1)  n$$ This is a very useful function in Counting problems. This function also pops up in the exponential function: $$e^x = 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...$$ Setting this aside, we think: can we 'extend' this function to all real number? Surely, even if we were to come up with such a function, what would it mean to take this new 'factorial' of some real number? Well, let's take a look at one important property of the factorial function: $$n! = n(n-1)!$$ and this is true for all natural numbers $n$. So, even if we came up with this homologous function, we...
Post a Comment
Read more

Is this seat taken?

100 people are waiting to board a plane. The first person’s ticket says Seat 1; the second person in line has a ticket that says Seat 2, and so on until the 100th person, whose ticket says Seat 100. The first person ignores the fact that his ticket says Seat 1, and randomly chooses one of the hundred seats (note: he might randomly choose to sit in Seat 1). From this point on, the next 98 people will always sit in their assigned seats if possible; if their seat is taken, they will randomly choose one of the remaining seats (after the first person, the second person takes a seat; after the second person, the third person takes a seat, and so on). What is the probability the 100th person sits in Seat 100? This problem can be solved intuitively. The first step to solve this problem is to understand that the last person will either get his seat or the first person's seat. But why? If the 1st person chooses 1st seat itself, everybody gets their own seat (i.e., the last person will g...
Post a Comment
Read more