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