# A Counter-Example that the Scholz-Brauer Conjecture holds with Equality for all n

 Index/Indices Element(s) Number of Elements resp. Indices Pair of Indices to form First Element Left Summand of First Element Right Summand of First Element First Element is generated by Smallstep No Number of 1-bits in binary representation 0 1 1 - 1 1 2 1 (0,0) $$1$$ $$1$$ - 1 2,...,5 3,...,24 4 (1,0) $$2$$ $$1$$ 1 2 6,...,10 $$2 (2^4-1),...,2^5 (2^4-1)$$ 5 (5,3) $$24$$ $$6$$ 2 4 11 $$2^9-29$$ 1 (10,2) $$2^5 (2^4 - 1)$$ $$3$$ 3 6 12,...,15 $$2^{10}-61,...,2^3(2^{10}-61)$$ 4 (11,10)$$^*$$ $$2^9-29$$ $$2^5 (2^4-1)$$ - 6 16 $$2^{13}-5$$ 1 (15,11) $$2^3(2^{10}-61)$$ $$2^9-29$$ 4 12 17 $$2^{13}-3$$ 1 (16,1)$$^*$$ $$2^{13}-5$$ $$2$$ 5 12 18,...,28 $$2^3 (2^{11}-1),...,2^{13} (2^{11}-1)$$ 11 (17,16)$$^*$$ $$2^{13}-3$$ $$2^{13}-5$$ - 11 29 $$2^{24}-3$$ 1 (28,17) $$2^{13} (2^{11}-1)$$ $$2^{13}-3$$ 6 23 30 $$2^{24}-1$$ 1 (29,1) $$2^{24}-3$$ $$2$$ 7 24 31,...,53 $$2^2 (2^{23}-1),...,2^{24} (2^{23}-1)$$ 23 (30,29)$$^*$$ $$2^{24}-1$$ $$2^{24}-3$$ - 23 54,...,78 $$2^{47}-1,...,2^{24} (2^{47}-1)$$ 25 (53,30) $$2^{24} (2^{23}-1)$$ $$2^{24}-1$$ 8 47 79,...,126 $$2^{71}-1,...,2^{47} (2^{71}-1)$$ 48 (78,30) $$2^{24} (2^{47}-1)$$ $$2^{24}-1$$ 9 71 127,...,245 $$2^{118}-1,...,2^{118} (2^{118}-1)$$ 119 (126,54) $$2^{47} (2^{71}-1)$$ $$2^{47}-1$$ 10 118 246,...,482 $$2^{236}-1,...,2^{236} (2^{236}-1)$$ 237 (245,127) $$2^{118} (2^{118}-1)$$ $$2^{118}-1$$ 11 236 483,...,955 $$2^{472}-1,...,2^{472} (2^{472}-1)$$ 473 (482,246) $$2^{236} (2^{236}-1)$$ $$2^{236}-1$$ 12 472 956,...,1900 $$2^{944}-1,...,2^{944} (2^{944}-1)$$ 945 (955,483) $$2^{472} (2^{472}-1)$$ $$2^{472}-1$$ 13 944 1901,...,3789 $$2^{1888}-1,...,2^{1888} (2^{1888}-1)$$ 1889 (1900,956) $$2^{944} (2^{944}-1)$$ $$2^{944}-1$$ 14 1888 3790,...,7566 $$2^{3776}-1,...,2^{3776} (2^{3776}-1)$$ 3777 (3789,1901) $$2^{1888} (2^{1888}-1)$$ $$2^{1888}-1$$ 15 3776 7567,...,7581 $$2^{7552}-1,...,2^{14} (2^{7552}-1)$$ 15 (7566,3790) $$2^{3776} (2^{3776}-1)$$ $$2^{3776}-1$$ 16 7552 7582,...,15145 $$2^3 (2^{7563}-1),...,2^{7566} (2^{7563}-1)$$ 7564 (7581,18) $$2^{14} (2^{7552}-1)$$ $$2^3 (2^{11}-1)$$ 17 7563 15146,...,30272 $$2^3 (2^{15126}-1),...,2^{15129} (2^{15126}-1)$$ 15127 (15145,7582) $$2^{7566} (2^{7563}-1)$$ $$2^3 (2^{7563}-1)$$ 18 15126 30273,...,60525 $$2^3 (2^{30252}-1),...,2^{30255} (2^{30252}-1)$$ 30253 (30272,15146) $$2^{15129} (2^{15126}-1)$$ $$2^3 (2^{15126}-1)$$ 19 30252 60526,...,121030 $$2^3 (2^{60504}-1),...,2^{60507} (2^{60504}-1)$$ 60505 (60525,30273) $$2^{30255} (2^{30252}-1)$$ $$2^3 (2^{30252}-1)$$ 20 60504 121031,...,242039 $$2^3 (2^{121008}-1),...,2^{121011} (2^{121008}-1)$$ 121009 (121030,60526) $$2^{60507} (2^{60504}-1)$$ $$2^3 (2^{60504}-1)$$ 21 121008 242040,...,484056 $$2^3 (2^{242016}-1),...,2^{242019} (2^{242016}-1)$$ 242017 (242039,121031) $$2^{121011} (2^{121008}-1)$$ $$2^3 (2^{121008}-1)$$ 22 242016 484057,...,968089 $$2^3 (2^{484032}-1),...,2^{484035} (2^{484032}-1)$$ 484033 (484056,242040) $$2^{242019} (2^{242016}-1)$$ $$2^3 (2^{242016}-1)$$ 23 484032 968090,...,1936154 $$2^3 (2^{968064}-1),...,2^{968067} (2^{968064}-1)$$ 968065 (968089,484057) $$2^{484035} (2^{484032}-1)$$ $$2^3 (2^{484032}-1)$$ 24 968064 1936155,...,3872283 $$2^3(2^{1936128}-1),...,2^{1936131} (2^{1936128}-1)$$ 1936129 (1936154,968090) $$2^{968067} (2^{968064}-1)$$ $$2^3 (2^{968064}-1)$$ 25 1936128 3872284,...,7744540 $$2^3(2^{3872256}-1),...,2^{3872259}(2^{3872256}-1)$$ 3872257 (3872283,1936155) $$2^{1936131} (2^{1936128}-1)$$ $$2^3 (2^{1936128}-1)$$ 26 3872256 7744541,...,15489053 $$2^3(2^{7744512}-1),...,2^{7744515}(2^{7744512}-1)$$ 7744513 (7744540,3872284) $$2^{3872259} (2^{3872256}-1)$$ $$2^3 (2^{3872256}-1)$$ 27 7744512 15489054,...,30978078 $$2^3(2^{15489024}-1),...,2^{15489027}(2^{15489024}-1)$$ 15489025 (15489053,7744541) $$2^{7744515} (2^{7744512}-1)$$ $$2^3 (2^{7744512}-1)$$ 28 15489024 30978079,...,30978147 $$2^3(2^{30978048}-1),...,2^{71}(2^{30978048}-1)$$ 69 (30978078,15489054) $$2^{15489027} (2^{15489024}-1)$$ $$2^3 (2^{15489024}-1)$$ 29 30978048 30978148 $$2^{30978119}-1$$ 1 (30978147,79) $$2^{71} (2^{30978048}-1)$$ $$2^{71}-1$$ 30 30978119
$$^*$$ means at least 1 carry occurs in binary addition; otherwise a missing asterisk means no carry occurs.

Thus we have $$l(2^{30978119}-1)\le 30978148<l(30978119)+30978119-1=31+30978119-1=30978149$$.

This counter-example is a star-chain for 230978119-1 which was constructed by Neill Clift on 1st July 2024.