# Online Encyclopedia

# Scholz conjecture

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In mathematics, the **Scholz conjecture** (sometimes called the **Scholz-Brauer conjecture** or the **Brauer-Scholz conjecture**) is a conjecture from 1937 stating that

*l*(2^{n}−1) ≤*n*− 1 +*l*(*n*)

where *l*(*n*) is the shortest addition chain producing *n*. It has been proved for many cases, but in general remains open.

As an example, *l*(5)=3 (since 1+1=2, 2+2=4, 4+1=5, and there is no shorter chain) and *l*(31)=7 (since 1+1=2, 2+1=3, 3+3=6, 6+6=12, 12+12=24, 24+6=30, 30+1=31, and there is no shorter chain), so

*l*(2^{5}−1) = 5−1+*l*(5).

## External links

## References

- Scholz, A., "Jahresbericht"
*Deutsche Math. Vereingung*1937 pp. 41-42 - Brauer, A. T., "On addition chains"
*Bull. Amer. Math. Soc.*1939 pp. 637-739

Last updated: 10-24-2004 05:10:45