For any positive integer, , there exists a unique positive integer so that . We proceed by strong induction on . For , the property is true as are themselves Fibonacci numbers. Suppose and that every integer less than is a sum of distinct Fibonacci numbers. Let be the largest positive integer such that . We first note that if then
giving us a contradiction. Hence and consequently the positive integer can be expressed as a sum of distinct Fibonacci numbers. Moreover, this sum does not contain the term as . Hence, is a sum of distinct Fibonacci numbers and Hogatt’s theorem is proved by induction.
|Date of creation||2013-03-22 13:43:24|
|Last modified on||2013-03-22 13:43:24|
|Last modified by||mathcam (2727)|