Mason’s theorem? Probably not, but cool nonetheless
I don’t know who worked out this particular proof first, but I doubt it was me. Still, I found myself working through some of the odd properties you can get with a number of the form b^n-1, and stumbled across the following proof:
Now, I noticed a couple of neat things I could derive from this. For one, any number that can be expressed as b^n - 1 where b and n are integers greater than 1 can always be evenly divided by b-1. And from that I can prove that any number that can be expressed as b^n - 1 can only be a prime if ‘b’ is 2.
Another thing I can show is that if ‘n’ is even, then b^n - 1 will be evenly divisible by b+1. The proof of that is based on simply replacing ‘n’ with ‘2n’ in the above proof, and then recognizing that b^2-1 is evenly divisible by b+1.
Ok, so maybe not as cool as the lottery ticket exercise, but I had fun working it out.
Filed under: proofs