Let the minimal length of an addition chain for a number
be denoted .
Then the Scholz conjecture, also called the Scholz-Brauer conjecture or Brauer-Scholz
conjecture, states that
The conjecture has been proven for a variety of special cases but not in general.
Brauer, A. T. "On Addition Chains." Bull. Amer. Math. Soc.45, 637-739, 1939.Gioia, A. A. and
Subbarao, M. V. "The Scholz-Brauer Problem in Addition Chains. II."
Congr. Numer.22, 251-274, 1978.Gioia, A. A.; Subbarao,
M. V.; and Sugunamma, M. "The Scholz-Brauer Problem in Addition Chains."
Duke Math. J.29, 481-487, 1962.Guy, R. K. Unsolved
Problems in Number Theory, 3rd ed. New York: Springer-Verlag, p. 169,
2004.Scholz, A. "Aufgabe 253." Jahresber. deutsche Math.-Verein.
II47, 41-42, 1937.Utz, W. R. "A Note on the Scholz-Brauer
Problem in Addition Chains." Proc. Amer. Math. Soc.4, 462-463,
1953.
be denoted 
.
Then the Scholz conjecture, also called the Scholz-Brauer conjecture or Brauer-Scholz
conjecture, states that
