Negabinary Numbers

Number Theory Level 5

\[\Large 20_{10} = 10100_2 = 10100_{-2}\]

Today, modern mathematics uses what is known as the decimal number system to write out large numbers. This means that when we write our numbers, we do so by thinking in powers of \(10\). For example, the number \(234\) is really \(2\times10^2+3\times10^1+4\times10^0\).

While the number \(10\) is the norm used around the world, numbers can really be written in any base. Take our example of \(234\) from above. It can be written as\(4\times7^2+5\times7^1+3\times7^0\), meaning that it's base \(7\) representation is \(453_7\). The subscript \(7\) denotes that the number is not in base 10, as to avoid confusion.

Not only can numbers be written in positive bases, but in negative bases as well! The example that can be seen at the top of the problem shows the conversion of the number \(20\) to Base \(2\), known as binary, and Base \(-2\), known as negabinary. As you can see, its representation is the same in both bases, but this is not the case with all numbers. For example, \(7=111_2=11011_{-2}\). How many positive integers \(n<1000\) have the same representation in both binary and negabinary?


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