<Factorial Compression> - All positive integers can be represented with a sum of factorials of positive integers. This is easily provable: -> All Positive Integers can be represented as a sum of many 1s (32 is 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1). -> 1! = 1 -> 1! is a factorial of a positive integer -> All Positive Integers can be represented as a sum of factorials of positive integers. - Because files are just binary code, you can turn them into a very large positive integer (with a maximum size of 2^(bytes*8)) - These numbers become very large very quickly. - However, factorials also grow very quickly, but until a number has its factorial taken, it remains relatively small. - Since any positive integer can be represented as a sum of these small numbers that can turn very large, perhaps they can be used as a compression method. - Here is a likely inefficient but effective method to compress files in this way, theoretically 1. Take the number representation F of some large file which we wish to compress. 2. Set N to 1. 3. If N! < F 3.1. Set N to N+1 3.2 Go to step 3. 4. Add N to the list of factorials 5. If N! is equal to F, stop. 6. Set F to F-N! 7. Go to step 2.