# Liouville Numbers

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, not only are there infinitely many, there uncountably many transcendental numbers, as they are called. This may be seen by noting there are only countably many polynomials with integer...

# Irrationality by Infinite Descent

The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the divisibility of the integers. (It is often covered in calculus courses and begins by assuming Sqrt=x/y where x/y is in smallest terms, then concludes that...

# 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 = zn, for integer powers n greater than 2? The French jurist and mathematician Pierre de Fermat claimed the answer was “no”, and in 1637 scribbled in the...

# 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 is an arithmetic progression. Question: does every increasing sequence of integers have to contain an arithmetic progression in it? For instance, the sequence of primes...

# Riemann Hypothesis

If you know about complex numbers, you will be able to appreciate one of the great unsolved problems of our time. The Riemann zeta function is defined by Zeta(z) = SUMk=1 to infinity (1/kz) . This is the harmonic series for z=1 and Sums of Reciprocal Powers if you set...

# Lucas’ Theorem

Lucas’ Theorem: If p is a prime number, and N has base p representation (aj,…,a1,a0) and k has base p representation (bj,…,b1,b0), then (N CHOOSE k) is congruent [mod p] to(aj CHOOSE bj)…(a1 CHOOSE b1)(a0 CHOOSE b0). Example: Let N = 588, k = 277, p = 5. N...

# Rational Irrational Power

If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt to the power 2 and you’ll get 2. But what happens if you raise an irrational number to an irrational power? Can this ever be rational?...

# 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 of Pascal’s Triangle?” For rows 0, 1, …, 20, we count: row N: 0 1 2 3 4 5 6 7 8 9 10 11...