- #1
Oxymoron
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When I write out the decimal expansion of [itex]1/p[/itex] where [itex]p[/itex] is a prime, it is always a recurring decimal with a period [itex]per(p)[/itex]. I was thinking why inverting a prime number should always give a recurring decimal but could not think of a reason other than it has to be something to do with our base 10 system of counting.
Some prime reciprocals, such as 1/7, 1/17, and 1/19 have periods which is one less that the prime. That is, [itex]per(p) = p-1[/itex] for some primes. And in fact, for all primes [itex]p[/itex], [itex]per(1/p)\,|\,(p-1)[/itex]. These are all interesting but there is something which I can't seem to explain.
Take the prime 11. Then the period of 1/11 is 2. Take some other prime p. Then
[tex]per\left(\frac{1}{11}\times\frac{1}{7}\right) = 2per\left(\frac{1}{p}\right)[/tex]
so long as [itex]p < 31[/itex].
After looking at 1/11, I started looking at other prime reciprocals, hoping to find other relationships between the periods of 1/p, 1/q and 1/pq. Unfortunately, the only thing I could find was that
[tex]per\left(\frac{1}{p}\times\frac{1}{q}\right) = \infty[/tex]
whenever [itex]p=q[/itex] and the period of 1/p is odd.
I was hoping for a nice clean relationship between 1/p,1/q, and 1/pq but I could not find one at all. So I tried checking the web and again, no one seems to have found any relationships. So I came here. If anyone has seen, or done any work on comparing periods of different prime reciprocals, could you please share your knowledge.
Cheers.
Some prime reciprocals, such as 1/7, 1/17, and 1/19 have periods which is one less that the prime. That is, [itex]per(p) = p-1[/itex] for some primes. And in fact, for all primes [itex]p[/itex], [itex]per(1/p)\,|\,(p-1)[/itex]. These are all interesting but there is something which I can't seem to explain.
Take the prime 11. Then the period of 1/11 is 2. Take some other prime p. Then
[tex]per\left(\frac{1}{11}\times\frac{1}{7}\right) = 2per\left(\frac{1}{p}\right)[/tex]
so long as [itex]p < 31[/itex].
After looking at 1/11, I started looking at other prime reciprocals, hoping to find other relationships between the periods of 1/p, 1/q and 1/pq. Unfortunately, the only thing I could find was that
[tex]per\left(\frac{1}{p}\times\frac{1}{q}\right) = \infty[/tex]
whenever [itex]p=q[/itex] and the period of 1/p is odd.
I was hoping for a nice clean relationship between 1/p,1/q, and 1/pq but I could not find one at all. So I tried checking the web and again, no one seems to have found any relationships. So I came here. If anyone has seen, or done any work on comparing periods of different prime reciprocals, could you please share your knowledge.
Cheers.
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