Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2008 / B / 5 / ii

The expansion is

- \frac{13}{8} = 1 + 3 + 2 \cdot 3^{2} + 3^{3} + 2 \cdot 3^{4} + 3^{5} + 2 \cdot 3^{6} + \dots = 1, \overline{12} in base 3.

(1)

The notation has the units digit before the comma and successive positive powers afterwards, as defined in the source. To check it, the geometric series converges in the 3-adic norm and its sum is

1 + (3 + 2 \cdot 3^{2}) / (1 - 3^{2}) = 1 - 21/8 = - 13/8

For the general repeating block,

∣ p^{2} ∣_{p} < 1

gives

a_{0}, \overline{a_{1} a_{2}} = a_{0} + \frac{a _{1} p + a _{2} p ^{2}}{1 - p ^{2}} = \frac{a _{0} ( 1 - p ^{2} ) + a _{1} p + a _{2} p ^{2}}{1 - p ^{2}} .

(2)

Both numerator and denominator are integers, and the denominator is a

p

-adic unit.

 Ancestors (12)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version

Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2008 / B / 5 / ii

Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1) New discussion

 Articles by others on the same topic (0)

 Discussion (0) New discussion