How can I commercially profit from the discovery of new algorithms?

It is not true that you can not patent your algorithm.

There are specific legal tricks that turns a pure algorithm into a patentable system in the US. The answer to your question is to talk to a patent lawyer who understands this.

You need to write a product around it. Look at google: they have secret algorithms too. They cannot have a patent on this (or at least not in the EU). But they can protect a product. So the algorithm is a secret and the product is what they sell.

Another option is to write a book/white paper about it and become famous.

You must realize that when you make such an extravagant claim, people will not believe you.

Your first step should be to talk to a recognized expert in the field of your invention. If you can convince one expert that you are for real, then doors will open. It's called social proof.

Some inventions are really hard to monetize. Einstein didn't exactly get rich from discovering relativity.

In your case, you may want to think about who would benefit from your algorithm, and license it to them. This way, the algorithm would remain secret, and only select companies would pay you lots of money for the code. You'd protect your IP as a trade secret, not a patent or copyright.

Yes, you can patent your Sorting Algorithm. See this patent as an example.. http://www.google.com/patents?id=GrTJAAAAEBAJ&pg=PA22&dq=sorting+algorithm&hl=en&sa=X&ei=5a0wT87lFuitsAKy8tTZBg&ved=0CDIQ6AEwAA#v=onepage&q=sorting%20algorithm&f=false

Do you know about the Provisional Patent Application? It only costs $125 and was designed by Congress to give individual inventors a chance at bringing their inventive ideas to market easily. It is super easy to do.

Intellectual Ventures buys patents, so if you can patent an implementation of it... http://www.intellectualventures.com/Home.aspx

Of course, there's no guarantee they won't just give the algorithm to Microsoft under the covers before any deal is made on paper.

Not a fan of IV, as you can see. You could make it open source. OR... Build a startup around it, since this site is about startups.

You cannot patent an equation, but an algorithm it's something else. There are ways this can be done, including Europe. I know because I've been to patent offices in Europe.

You should patent it first. If you have the money, worldwide. Then you could: (a) build an application that uses the algorithm and sell the software, or (b) go to big companies and show them your software or patented algorithm and see if they are interested in buying.

Making a software could also include making an API and give access to it in SaaS plataform. Just thinking out loud.

Best of lucks.

You can copyright the implementation of an algorithm, I.e. the code base . You can patent the solution to problem that is industrially applicable and includes an invention step. As such an algorithm may be patented under very strict terms, I.e. it is not a variation or an improvement of an already existing algorithm. This is very difficult to justify. You cannot patent or copyright an equation or a mathematical solution to a problem, because an equation by itself is not industrially applicable. Beware, that patents are granted by patent offices residing in different countries. However, a patent is an archive that you have reported the solution to a problem. Enforcing patent claims is a different story, and a patent being accepted in one country does really verify it's validity in another. In fact patent claims are enforced in each different country where there is an infringement. My advise, make it public, be famous, or keep it a secret.