Hmm, actually, instead of the prime P that I had in mind (in my comment earlier today) the prime you construct, with respect to which all the square roots exist except one — that prime really does work in my proof. Which then is seen to be a somewhat disguised version of your proof.

]]>Was that proof valid? I think it all is ok except I am worried about the “then” clause inside the sentence “Now…”.

]]>The argument is more specifically that if they can be solved in a finite extension of , then they can be solved in the ring of integers of that finite extension, and so they can be solved in the ring of integers of that finite extension .

]]>Given any n tuple of positive integers {a_i} find k such that {(a_i^k)+1} all have a common factor.

and I solved it with that conjecture in your post. I had the exact same proof for that conjecture as well. I really like this post of yours by the way.

]]>Notice that the trace of an algebraic integer is an algebraic integer. A quick computation shows that . So, if is an algebraic integer, then , and are all integers.

Now, you just need to do local analyses at the primes 2 and 3. I have reasonably elegant ways of doing these, but nothing super-slick.

]]>