How can you compute to 14 digits of accuracy within the domain ?

# Floating point quirks

In this post, we’ll explore a scenario where the non-~~commutativity~~associativity of floating point arithmetic can lead us into trouble.

Let and . Write a computer program to compute this sum.

# Solving first order PDEs with Characteristics

Use the method of characteristics to solve the PDE system

# Power Reduction in Congruences

Suppose you have integers a,b that are relatively prime to m such that

then

# Even Pascals

Let , where k is an integer. Prove that for any positive integer n the number

is divisible by .

# Sequences in Sequences

Define the sequence recursively by and

Find an explicit formula for in terms of n.

# More Linear Recurrences

Let be defined by the recurrence relation , with . Show that the expression depends only on b and , but not on a.

# Almost Linear

Find the general term of the sequence given by , and

# Polynomial Divisors

Let . Show that for any positive integers there exists unique numbers such that the polynomial is divisible by p(x)

# Analytical Fibonacci

We derive Binet’s equation for the n^{th} Fibonacci number as