Recently I stepped on a series similar to 1 / 1+x \ 1 3 1 5 1 2k+1 - log | --- |=x+- x +- x +...+---- x +... 2 \ 1-x / 3 5 2k+1 The relation is / 2 \ 2 3 5 6 1 | 1+3x+3x | 3 5 3 7 3 11 3 13 - log | -------- |=x - -- x - -- x + -- x + -- x +...= 6 | 2 | 5 7 11 13 - \ 1-3x+3x / / 6k+0 6k+2 6k+3 6k+5 \ \~~ oo| 3 12k+1 3 12k+5 3 12k+7 3 12k+11 | = > | + ----- x - ----- x - ----- x + ------ x | /__ k=0\ 12k+1 12k+5 12k+7 12k+11 / Alternative form: / +-+ 2 \ 5 7 11 13 1 | 1+\|3 x+x | x x x x ------ log | ----------- |=x - -- - -- + --- + --- +...= +-+ | +-+ 2 | 5 7 11 13 - 2 \|3 \ 1-\|3 x+x / / 12k+1 12k+5 12k+7 12k+11 \ \~~ oo| x x x x | = > | + ------ - ------ - ------ + ------- | /__ k=0\ 12k+1 12k+5 12k+7 12k+11 / Seeking for relations of that type, I spotted the following, given in D.H.Bailey, R.E.Crandall: "On the Random Character of Fundamental Constant Expansions": / 2 \ 2 4 5 / 3k+1 3k+2 \ 1 | 1+x+x | x x x \~~ oo| x x | - log | ------- |=x + -- + -- + -- +... = > | + ----- + ----- | 3 | 2 | 2 4 5 - /__ k=0\ 3k+1 3k+2 / \ 1-2x+x / I could not any further relation of that type. -- p=2^q-1 prime <== q>2, cosh(2^(q-2)*log(2+sqrt(3)))%p=0 Life is hard and then you die.