Hello, the actual number is : 10766857856369919884615389387026267/5193981023518027157495786850488117+3588952618789973294871796462342089/5193981023518027157495786850488117*5^(1/2) which is a/b + c*gr/d, a,b,c,d being integers, gr = golden ratio. of course, one thing I made was a program to check various exponents and values of n and k (previous expression), the maximum seems to be when n is small. ... but not all the time. now we expect the expression to have a big initial term, actually in that example, 3 is the first term, and then the term : 83364870763649235403921261388869364666045817819140268784224747492762 appears 3 times in the first 54293 terms of the continued fraction. We all know that quadratic irrationals have a periodic continued fraction, and it is easy to show that. What is surprising is the length of the period of that example and the height of the maximal term. example : sqrt(3442321), it has 3710 as maximal value and the period occurs 46 times over 100000 terms, I just took a random example. this is strange, the first term of the expression is 10766857856369919884615389387026267/5193981023518027157495786850488117 a rational which has a FINITE c.f. the second term : 3588952618789973294871796462342089/5193981023518027157495786850488117*5^(1/2) is periodic (in the c.f. ) and the maximum is 16672974152729847080784252277773872933209163563828053756844949498552. bonne journée, Simon Plouffe