We considered the pentagonal numbers which begin : That is : for Now consider the following sequence : which begins at and has a common difference of between the terms. For any arithmetic sequence, we [more…]
Hypsicles of Alexandria (c.190 – c.120 B.C.) gave the following definition of polygonal numbers : If there are as many numbers as we please beginning from 1 and increasing by the same common difference, then, [more…]
One sometimes gets very interesting results when one turns divergent sums upside down. The Basel problem was asked by Pietro Mengoli in . He asked what is the exact summation of the reciprocals of the [more…]
If you sent all the gifts in the Twelve Days of Christmas, then how many would you send? The gifts for each day are the triangular numbers : We have done these before : In [more…]
The Möbius function is denoted by , where is the Greek letter mu. Depending on the factorisation of , the function takes the values . The last says that if is even, then and if [more…]
Some notation and definitions. is a finite group. is the order (size) of a group. is a finite set. is the order (size) of a set. Group action : For an action of the group [more…]
A proof of Fermat’s Little Theorem using necklaces. , where is prime and is any integer with gcd. Multiply both sides of the congruence by : Which gives : Subtract from both sides : [more…]
It is the 7th May 1945 and on this spring morning in the Bavarian mountains, a few days after the death of Hitler, the soldiers of the 44th infantry division that are manning a checkpoint [more…]
There are lots of technical problems in designing a liquid fuelled rocket; such as how do you light it, relight it, how do you stop combustion instability shaking the rocket to pieces, how do you [more…]
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