Here is a numerical curiosity: 588^2 + 2353^2 = 5882353 the curious thing is that 1/17 = 0.05882353 (rounded to 7 digits). ... are there other known results in this direction?
Yes. If A^2+1 divides B^A-A, then in base B, 1/(A^2+1) has that curious property. Examples: 3^2+1 divides 7^3-3, so in base 7: 1/(3^2+1) ~= .046_205, and 46^2+205^2=46205. 4^2+1 divides 6^4-4, so in base 6: 1/(4^2+1) ~= .0204_1225, and 204^2+1225^2=2041225. 5^2+1 divides 5^5-5, so in base 5: 1/(10^2+1) ~= .00440_04401, and 440^2+4401^2=44004401. 7^2+1 divides 43^7-7, so in base 43: 1/(7^2+1) ~= .(00)(36)(42)(06)(00)(36)(42)_ (06)(00)(36)(42)(06)(00)(37), and this one works, too. In each case, A says how many digits to take, and you round upward. I don't know of another base-ten instance. -- Don Reble djr@nk.ca