Before we can start to solve some counting problems we need to know a little of combinatorics’ notation. Luckily, a few simple rules should get us started. Don’t worry if it doesn’t make sense, that [more…]
Mathematics is not everyone’s cup of tea – fair enough. The Monday morning ‘double maffs’ articles by our resident mathematician, OldTrout, can present some GP’ers with a feeling of bewilderment. Myself included. I’m no mathematician. [more…]
We touched on quadratic residues when discussing Pythagorean triples. We relied upon the quadratic residues and non-residues in modulo for our proof that and took opposite parities. These residues have important results in encryption, integer [more…]
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…]