My younger son and I have reached the end of Art of Problem Solving’s Introduction to Number Theory book and I have a strange problem – he loved it.
I mean LOVED it. When we ran out of problem to work on yesterday, he was almost in tears because Khan Academy didn’t have any problems on modular inverses. Ha!
Anyway, he’s begging to do more, but I’m not aware of any other book on number theory that could even pretend to be accessible to young kids.
I’d planned on diving into the 2nd half of Art of Problem Solving’s Prealgebra book when we finished this book, but it isn’t as though I’m in any sort of hurry. I’d love to spend another month or two talking through more number theory ideas with him, but I really don’t know where to look. A quick check for number theory resources on Amazon, for example, brings up stuff like Hardy and Wright. Not exactly what I’m looking for 🙂
Any ideas on where to look for some fun and basic number theory?
Mike's Math Page 
Number theory soon develops into “density of primes”, log functions, and unsolved problems about quantity of prime pairs (infinite or not), and stuff like “can every number be expressed as the sum of four squares”.
Since he is into modular arithmetic there are two directions to go. One is computer programming, especially with mod and div functions, for defining and using circular arrays as buffers. the other one, more mathematical, is to move towards analysing the structure of the addition table and the multiplication table (separately), as “finite groups”, to use the technical jargon. Then to look at transformation groups, for example rotations through 90 or 72 degrees, where the combining rule is “do one move, then another, what’s the result”.
This can get quite interesting, and gets away from “it’s all about numbers”.
Three rich areas for exploration that will be accessible for your son:
(1) continued fractions. Launch question: we know about fractions, what if we tried this little extension. What do we get?
(2) quadratic residues. We have thought about linear equations in modular arithmetic. What about quadratic equations, anything interesting there?
(3) diophantine equations: maybe work on pythagorean triples helps kick-off this?
Davenport’s Higher Arithmetic may be useful to you.
In all cases, I would encourage a lot of numerical investigation to look for patterns and generate conjectures. FWIW, that is the spirit of the Ross/PROMYS tradition.