Find Any Integer Pair That Needs Exactly K Euclidean Divisions to Make B Zero
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Find Any Integer Pair That Needs Exactly K Euclidean Divisions to Make B Zero
Assume two integers and () that form an ordered integer pair . We can perform a Euclidean division on it, which will make it . If we keep performing this division, then will eventually be . We define as the number of Euclidean divisions needed to make equal .
Example: → → → → . We perform divisions until equals , so .
With the value of specified (), how to find any ordered integer pair () that needs exactly Euclidean division to make equal ? If there does not exist such a pair, report it.
I am also curious about one question: it is all known that the Euclidean algorithm is fast, but the speed of the algorithm is determined by , so may be infinitely large?
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