Re: Square roots

In article <BF02F8F3.5E6A%jean-claude.arbaut@xxxxxxxxxxx>,
Jean-Claude Arbaut <jean-claude.arbaut@xxxxxxxxxxx> wrote:

bischar <bisch_a_r@xxxxxxxx> wrote:

> 1/7 = 0.142857142857...
>
> 3/7 = 0.428571428571...

etc.

>But, what did you mean ? Everybody knows that every rationnal has a periodic
>decimal expansion. What else ?

I assumed that bishar was pointing out that not only do 1/7, 2/7, etc.
all have periodic decimal expansions, but that the decimal expansion for
each is obtained by shifting over the decimal digits for 1/7.
(Surely that is why he listed 3/7 immediately after 1/7, rather than
going next to 2/7.)

But of course this is nearly as trivial as the existence of a periodic
expansion. "Shifting digits" corresponds to multiplying by 10, so as
one shifts digits repeatedly, one essentially computes (the decimal
expansions for 1/7 of) the residues of the powers of 10 modulo 7.
So the fact that all the expansions of n/7 can be obtained by
shifting digits is equivalent to the fact that 10 (i.e. 3) is a primitive
root modulo 7. For some primes (e.g. 7, 17, ... ) 10 is indeed a
primitive root mod p, and for other primes (3, 11, 13, 31, ...) it isn't.
So the "amazing pattern" seen for sevenths is also seen for seventeenths
1/17 = .05882352941176470588235294117647058823529411764705...
10/17 = .58823529411764705882352941176470588235294117647059...
15/17 = .88235294117647058823529411764705882352941176470588...
14/17 = .82352941176470588235294117647058823529411764705882...
4/17 = .23529411764705882352941176470588235294117647058824...
(etc.) but not for elevenths.

It is conjectured that 10 is a primitive root for infinitely many
primes but this is not known.

dave
.

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