Hacker's Delight

10-7. Miscellaneous Topics

Theorem DC2. The least multiplier m is odd if p is not forced to equal W.

Proof. Assume that Equations (1a) and (1b) are satisfied with least (not forced) integer p, and m even. Then clearly m could be divided by 2 and p could be decreased by 1, and (1a) and (1b) would still be satisfied. This contradicts the assumption that p is minimal.

Uniqueness

The magic number for a given divisor is sometimes unique (e.g., for W = 32, d = 7), but often it is not. In fact, experimentation indicates that it is usually not unique. For example, for W = 32, d = 6, there are four magic numbers:

However, there is the following uniqueness ...