Cool web site - 460 ARITHMETIC 4.6.3 As an example of an
460 ARITHMETIC 4.6.3 As an example of an lo-chain (certainly not a minimum one), consider 1 2 4 5,&10,12,l8; -, -, -, (51) it is easy to verify that the difference between each element and the previous underlined element is in the chain. We let lo(n) denote the minimum length of an IO-chain for n. Clearly l(n) 5 1 (n) 5 l*(n). The chain constructed in Theorem F is an lo-chain (see exercise 22); hence we have lo(n) < 1*( n ) f or certain n. It is not known whether or not l(n) = 1 (n) in all cases; if this equation were true, the Scholz-Brauer conjecture would be settled, because of another theorem due to Hansen: Theorem G. 1 (2 -1) 2 n -1 + I (n). Proof. Let 1 = ao, al, . . , a, = n be an lo-chain of minimum length for n, andlet 1 =bo, bi, . . . . bt = n be the subsequence of underlined elements. (We may assume that n is underlined.) Then we can get an lo-chain for 2n -1 as follows: a) Include the 1 (n) + 1 numbers 2at - 1, for 0 2 i 5 T, underlined if and only if a, is underlined. b) Include the numbers 2i(2b J--l),forO