?? powers of 7 mod 10 ...

1,7,9,3

?? powers of 10 mod 7 ...

1,3,2,6,4,5

?? ... so what's the relationship to 1,4,2,8,5,7 ??

hmm, at least i see why i tried the first one first ... because clearly 1,4,2,8,5,7 live mod 10 rather than mod 7 ... but so what's going on here ??? _is_ it some sort of "reciprocity" ?? .... ???? .....

obviously very "simple" stuff going on here, but never learned about it, as far as i can remember ...



7 goes into 10 once, with remainder 3 .... ??? hmm, maybe 3,2,6,4,5,1 are remainders corresponding to truncated quotients 1,14,142,1428,14285,142857 ???

10/7,100/7,1000/7,10000/7,100000/7,1000000/7

?? somehow you're done when you get to a remainder of 1 ... 999999/7 = 142857 ....

?? harmonic series ... ??? .....

?? remainders live modulo 7 .... ?? but then look at the truncated quotients mod 10 .... ??? remainders wrt 10 of truncated quotients wrt 7 .... ??? ..... ???? ..... ??? ....

?? analogy between polynomials over z/n (especially with n prime ?? ...) and integers .... ??? but .... actual map from former to latter .... ????? parody of evaluation at zero .... ????? ...... ?????? .... ??? map other way .... ???? ..... ???? ..... ?? mediated by .... polynomials over z .... ???? ......