Wednesday, 23 March 2016

elementary number theory - Do such unique representations of positive integers exist?




It is well known that every positive integer n>0 can be represented uniquely in the form
n=2k(2m+1),


for positive integers k,m0. Does there exist one or more constants c>1 such that
2k(2m+c)

is a unique representation for positive integers greater than some lower bound?


Answer




For c=3 the expression is unique, though you can not get a lot of numbers this way. For example, no power of 2 can be written this way.



To see that, for the numbers which can be expressed this way, the expression is unique: Just remark that 2k1(2m1+3)=2k2(2m2+3)k1=k22m1+3=2m2+3m1=m2



A similar argument goes through for any odd c. For even c the argument fails since we can not conclude that k1=k2.


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