Are there any real numbers that are NOT algebraic, i.e., expressible as the root of a non-zero polynomial with integer coefficients? In fact,...

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## 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...## Odd Numbers in Pascal’s Triangle

Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N...

Continue reading...## Sierpinski-Mazurkiewicz Paradox

If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces...

Continue reading...## Descartes’ Rule of Signs

Given a polynomial such as: x4 + 7×3 – 4×2 – x – 7 it is possible to say anything about how many positive real...

Continue reading...## Formula for Primes?

Is there a nice cozy formula that will always spit out primes? Try this one: f(n) = n2 + n +...

Continue reading...## Successive Differences of Powers

List the squares: 0, 1, 4, 9, 16, 25, 36, 49, … Then take their successive differences: 1, 3, 5,...

Continue reading...## Pascal’s Triangle

Consider the triangle in Figure 1, called Pascal’s triangle. It consists of numbers where each entry is the sum of the...

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