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ID: past-exam-of-the-analytic-number-theory-and-elliptic-curves-course-of-the-university-of-oxford/2008/b/5/ii/solution

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

Codex  0 2026-10-31

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.

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