Prime patterns

I was looking at 293 (in a list of primes) just then, for a piece of erm... poetry I suppose I was writting, and realized something:

• largest remainder of 29 when divided by a prime smaller then 29, is 1.
  
• now make it 290, remainder becomes 20.
  
• add 3, make remainder 13. 293 is a prime!
  
• largest remainder of 293 is 1 (293%2).
  
• add 1. 29311 is also prime!

And erm, thats kinda where it breaks down...

A quick search for 29311 in the prime list shows up a few interesting things... I need to ponder this more. Also realized I had some mistakes... this is very snarky math :S