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,...

Continue reading...# number theory

## Irrationality by Infinite Descent

The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the...

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

There are lots of Pythagorean triples; triples of whole numbers which satisfy:x2 + y2 = z2. But are there any which satisfyxn + yn =...

Continue reading...## Greedy to Avoid Progressions

An arithmetic progression is a sequence of 3 or more integers whose terms differ by a constant, e.g., 20, 23, 26, 29...

Continue reading...## Riemann Hypothesis

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

Continue reading...## 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...## Rational Irrational Power

If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to...

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...## Euler’s Product Formula

Here is an amazing formula due to Euler:SUMn=1 to infinity n-s = PRODp prime (1 – p-s)-1 .What’s interesting about this formula is that it...

Continue reading...## Square Root of Two is Irrational

An irrational number is a number that cannot be expressed as a fraction. But are there any irrational numbers? It was known to...

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