Scholz Conjecture

Let the minimal length of an Addition Chain for a number $n$ be denoted $l(n)$. Then the Scholz conjecture states that

$$l(2^n-1)\leq n-1+l(n).$$

The conjecture has been proven for a variety of special cases but not in general.

See also Addition Chain

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 111, 1994.