### Why it isn’t a 50 : 50 chance (Revised)

Phil the Test Manager brought up an interesting point on probability the other day.  When playing in a bridge contract where you and your partner have eleven of the trump card between you, what are [more…]

### Euler’s Number

“To express remember to memorise a sentence to memorise.” This mathematical constant is a rate of growth.  It was discovered by Jacob Bernouilli  whilst studying compound interest.  Euler referred to the constant as  and it [more…]

### The Bravais Lattices Song

By Walter F. Smith Sing to the tune of “I Am the Very Model of a Modern Major General” from “The Pirates of Penzance”.   If you have to fill a volume with a structure [more…]

### Fermat’s Little Theorem

Fermat was able to factorise large numbers, such as the above, long before the days of calculators and computers by making use of his little theorem.  One could try trial division by primes less than [more…]

### The Method of Infinite Descent – Fermat’s Favourite Proof

Fermat left only one proof. The area of a Pythagorean triangle is never a square number. Fermat wrote ,  “If the area of a right-angled triangle were a square, there would exist two biquadrates (fourth [more…]

### Regular Numbers and Plimpton 322

There are trigonometric arguments for interpretation of Plimpton 322 and a number-theoretic argument by Neugebauer.

### A Rooted Ternary Tree

There is another way of generating the primitive Pythagorean triples using matrices which was discovered by Berggren in 1934 and again by Barning in 1963.

### The Pythagorean Equation

We consider another quadratic Diophantine equation this time in three variables; namely, the Pythagorean Equation : It is called the Pythagorean Equation although the Ancient Egyptians and Babylonians certainly knew how to generate Pythagorean triples. [more…]

### Infinite Continued Fractions

As proved by Euler, the value of any infinite continued fraction is an irrational number.  Just as every finite continued fraction is a rational number, every infinite continued fraction represents an irrational number. We consider [more…]

### Simple Finite Continued Fractions

A finite continued fraction looks that this : there is an integer part and a nest of fractions. The integer part may be negative, zero or positive. If negative, then it is an improper negative [more…]

### The Silver Ratio

Just as there is a golden ratio,  , based on the Fibonacci numbers, there is a silver ratio, , based on the Pell numbers.   John Pell, . The continued fraction for is : The [more…]

### Standing on the Shoulders of Giants

I had always associated the phrase “If I have seen further, it is by standing on the shoulders of giants” with Sir Isaac Newton. (Letter to Hooke, ).  That is, we build our knowledge on [more…]

### Hilbert’s Paradox of the Grand Hotel

Mr. RotherhamPoofta visited the site the other day and, as is often the case, he asked me if he should book a room.  He is very sweet and kind and asks this of many people. [more…]

### Sophie Germain Primes and Cunningham Chains

A Sophie Germain prime is a prime number such that is also a prime number. That is, ,  where and are both prime.  For example, . The sequence begins Sophie Germain () explored these primes [more…]

### Perfect Numbers and Mersenne Primes

Among the many wonderful things contained in Euclid’s ‘Elements’ , written in the century , is Proposition of Book : ‘If as many numbers as we please beginning from a unit be set out continuously [more…]