## The Polynomials that Generate Prime Numbers

The search for primes started thousands of years ago. Mathematicians since antiquity tried to find ways to look for primes. They also searched for methods to test if a number is prime or not. Others tried to find polynomials to generate primes.

The Sieve of Eratosthenes

One of the ancient methods of listing prime numbers is the Sieve of Eratosthenes. The Sieve consists of a finite list of numbers, where the multiples of each number are crossed out starting from 2 and increasing each time the list is exhausted.  » Read more

## Are all fractions rational numbers?

No.

A rational number can be expressed in the form $\displaystyle\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. In other words, it is a fraction whose denominator is not zero, and both the denominator and numerator are integers.

Some fractions, however, may contain a numerator or denominator that is not an integer. Some examples of such fractions are

$\displaystyle\frac{\sqrt{3}}{2}$, $\displaystyle\frac{\pi}{4}$ and $\displaystyle\frac{e}{2}$.

A rational number may be represented in many ways, but it can always be expressed as a fraction. For instance, $10^{-1}$ is a rational number because we can express it as $\frac{1}{10}$. Also, the number $0.333 \cdots$, a repeating decimal, is  a rational number because we can also express it as fraction $\frac{1}{3}$.

## Guest Post: An Interesting Property of Prime Numbers

Although I have already discussed modulo division, I believe that this proof is beyond the reach of average high school students. To explain further, I made additional notes on Patrick’s proof . I hope these explanations would be able to help students who want to delve on the proof.

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I’ve got a prime number trick for you today.

1. Choose any prime number $p > 3$.
2. Square it.
3. Add 5.
4. Divide by 8.

Having no idea which prime number you chose, I can tell you this:

The remainder of your result is 6.  » Read more

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