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...

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 divisibility of the integers. (It is often covered in calculus courses and begins by assuming Sqrt[2]=x/y where x/y is in smallest terms, then concludes that...

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 = 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...

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 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...

Continue reading...# 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...

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 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...

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 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?...

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 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...

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 relates an expression involving all the positive integers to one involving just primes! And you can use it to prove there must be infinitely many primes....

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 the ancient Greeks that there were lengths that could not be expressed as a fraction. For instance, they could show that a right triangle whose...

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