If you know about complex numbers, you will be able to appreciate one of the great unsolved problems of our...

Continue reading...# prime

## Lucas’ Theorem

Lucas’ Theorem: If p is a prime number, and N has base p representation (aj,…,a1,a0) and k has base p...

Continue reading...## Sum of Cubes and Beyond

We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the...

Continue reading...## Sums of Two Squares

Which whole numbers are expressible as sums of two (integer) squares? Here’s a theorem that completely answers the question, due...

Continue reading...## All Numbers are Interesting

There are clearly many interesting whole numbers. For instance, 2 is the only even prime number, 3 is the first...

Continue reading...## Fibonacci GCD’s, please

Fibonacci numbers exhibit striking patterns. Here’s one that may not be so obvious, but is striking when you see it....

Continue reading...## Sum of Prime Reciprocals

It is a well-known fact that the harmonic series (the sum of the reciprocals of the natural numbers) diverges. But what about...

Continue reading...## How many Primes?

Are there infinitely many primes? We’ll give a proof, due to Euclid, to show that there must be infinitely many primes....

Continue reading...## Gaps in Primes

We know there are infinitely many primes, so are many interesting questions you can ask about the distribution of primes, i.e., how...

Continue reading...## Fermat’s Little Theorem

Fermat’s little theorem gives a condition that a prime must satisfy: Theorem. If P is a prime, then for any...

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