High Quality Content by WIKIPEDIA articles! In mathematics, the Scholz conjecture (sometimes called the Scholz-Brauer conjecture or the Brauer-Scholz conjecture) is a conjecture from 1937 stating that l(2n?1) ? n ? 1 + l(n) where l(n) is the length of 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(25?1) = 5?1+l(5). Simple number-theoretic investigation into the nature of the addition chain and the binary representation of a number allows us to prove this weaker inequality: l(2n?1) ? 2n ? 2. A proof reducing one of the ns to an l(n) has yet to be found.

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