Let’s take a look at an example of a situation where ambiguity occurs during the derivation of subtyping proofs in Java. In general, it’s conjectured that subtyping in Java is undecidable.

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

Use the method of characteristics to solve the PDE system

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

is divisible by .

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

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

We derive a general technique for solving full rank linear recursive sequences. Formally, a k^th linear recursive sequence is defined as

Find all primes p and q such that

Show that there are infinitely many primes of the form 4k - 1.

Suppose we define the function as the sum of the divisors of n

Express in terms of ‘s prime factorization

Determine the number of ordered pairs of positive integers such that their least common multiple

Let a and b be distinct integers such that . Show that

The sequence satisfies

for all nonnegative integers n,m and . If , determine

Find a closed form formula for the recurrence relation

Find a closed form formula for the sequence {1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,…}

If a positive integer multiple of 864 is chosen randomly, with each multiple having the same probability of being chosen, what is the probability that it is divisible by 1944?

This is a classic problem first solved by Euclid: Show that there are infinitely many prime numbers.

If p and q are primes, and x² – px + q = 0 has distinct positive integer roots, find p and q.

Let n be a positive integer. Prove that 3²ⁿ+1 is divisible by 2,but not by 4.

Let k be an even number. Is it possible to write 1 as the sum of the reciprocals of k odd integers?

Twenty bored students take turns walking down a hall that contains a row of closed lockers, numbered 1 to 20. The first student opens all the lockers; the second student closes all the lockers numbered 2, 4, 6, 8, 10, 12, 14,16, 18, 20; the third student operates on the lockers numbered 3, 6, 9, 12, 15, 18: if a locker was closed, he opens it, and if a locker was open, he closes it; and so on. How many lockers remain open after all the students finish?

1. Find out what 2.001⁵, 8.6^2/3, e^-0.15, and 1/1200 are approximately without using a calculator.

We use a really simple method that works well for polynomials and describe how we can determine how well our approximation does relative to the actual answer.

Binary representation of numbers and floating point precision arithmetic.

Once upon a time, I absolutely despised matlab. In my own spoiled preconceptions, I believed that matlab was designed without a single shred of care as to how ugly it is. Of course, I eventually had to finally write my first line of matlab and, upon discovering that I’m still in one piece, concluded that matlab wasn’t really that bad. Anyways, I’ve moved onto grander things now, things such as singing obnoxiously in public, browsing reddit for hours on end, and Python.

ISBN is an acronym for International Standard Book Number and is used as unique identifiers for published books. In this entry I will talk about how mathematics plays a role in the development and usage of the ISBN.

This series more or less mirrors the series of the same name for Python. It’s in my belief that the only way to learn the in and outs of a language is to learn by practice, and by that virtue, to practice as often as possible until you get the hang of the language.

Imagine that you are nine and you’re going to 3rd grade math class. Your teacher asks for the greatest common denominator between 10 and 15. Everyone else is just as stumped as you are, but being gifted with the incredible ability to program computers, you decide to write a program to do it for you.

Far too often, I hear people ask questions like “How am I suppose to know where to put ends for statement blocks?” or “Why can’t the computer just infer that I want this piece of code to do something else?”. These questions reflect something that the computer manufacturing industry have done quite successfully. They have succeeded in selling computers as smart general purpose platforms, which is an essential “feature” for about 99% of the computer users out there.

But programmers don’t usually work under these assumptions that a computer is smart. In order to fully understand programming, we need to start off with the fundamentals, that a computer is nothing more than a dumb calculator.

For years, my lack of confidence in my own creations have confined me to an online life of anonymity. Only recently have I discovered that other people aren’t all that scary. In fact, I realized that while people can suck sometimes, they’re usually quite decent.

As a lowly freshman struggling through college and life, I sympathize with the thousands of you out there who’re sometimes a little more than intimidated by the programming community. I am one such individual, and I hope that by the time I wear out this journal, I will be able to proudly claim that I am a passably mediocre programmer.