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Let b and k be p-adic numbers, we write b as a Power series in p with a given precison. Is It possible to write b^k as a Power series in k ?

Let b and k be p-adic numbers, we write b as a Power series in p with a given precison. Is It possible to write b^k as a Power series in k ?? An example : Let \gamma_1 and \gamma_2 be the 3-adic unit roots of the quadratic equations x^2+x+3=0 and x^2+2x+3=0 respectively. Let k be a 3-adic number such that v_3(k) >= 0. Let

C(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1

The problem is to show that v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))>=4. I know how to write \gamma_1 and \gamma_2 as a 3-adic power series with given precision, but have no idea how to work with the exponent.

Let b and k be p-adic numbers, we write b as a Power series in p with a given precison. Is It possible to write b^k as a Power series in k ? An example : Let \gamma_1 and \gamma_2 be the 3-adic unit roots of the quadratic equations x^2+x+3=0 and x^2+2x+3=0 respectively. Let k be a 3-adic number such that v_3(k) >= 0. Let

C(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1

The problem is to show that

v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))>=4. k^4))>=4 (1).

I know how to write \gamma_1 and \gamma_2 as a 3-adic power series with given precision, but have no idea how to work with the exponent.exponent. I`ve tried (1) for k integer between 1 and 100 and the valuation is zero or an integer greater than 3.

Let b and k be p-adic numbers, number, we write b as a Power series in p with a given precison. Is It possible to write b^k as a Power series in k , with k an integer ? An example : Let \gamma_1 and \gamma_2 be the 3-adic unit roots of the quadratic equations x^2+x+3=0 and x^2+2x+3=0 respectively. Let k be a 3-adic number such that v_3(k) >= 0. an integer. Let

C(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1

The problem is to show that

v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))>=4 (1).

I know how to write \gamma_1 and \gamma_2 as a 3-adic power series with given precision, but have no idea how to work with the exponent. I`ve tried (1) for k integer between 1 and 100 and the valuation is zero or an integer greater than 3.

Let b $b$ be p-adic number, we write b $b$ as a Power series in p $p$ with a given precison. Is It possible to write b^k $b^k$ as a Power series in k , $k$, with k $k$ an integer ? An example : Let \gamma_1 $\gamma_1$ and \gamma_2 $\gamma_2$ be the 3-adic unit roots of the quadratic equations x^2+x+3=0 $x^2+x+3=0$ and x^2+2x+3=0 $x^2+2x+3=0$ respectively. Let k $k$ be an integer. Let

C(k) $$C(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 11$$

The problem is to show that

v_3(c(k) $$v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))>=4 (1).(1).$$

I know how to write \gamma_1 $\gamma_1$ and \gamma_2 $\gamma_2$ as a 3-adic power series with given precision, but have no idea how to work with the exponent. I`ve I've tried (1) for k integer between 1 and 100 and the valuation is zero or an integer greater than 3.

Let $b$ be p-adic number, we write $b$ as a Power series in $p$ with a given precison. Is It possible to write $b^k$ as a Power series in $k$, with $k$ an integer ? An example : Let $\gamma_1$ and $\gamma_2$ be the 3-adic unit roots of the quadratic equations $x^2+x+3=0$ and $x^2+2x+3=0$ respectively. Let $k$ be an integer. Let

$$C(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1$$

The problem is to show that

$$v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))>=4 k^4))>=4\qquad (1).$$

I know how to write $\gamma_1$ and $\gamma_2$ as a 3-adic power series with given precision, but have no idea how to work with the exponent. I've tried (1) for k integer between 1 and 100 and the valuation is zero or an integer greater than 3.

Let $b$ be p-adic number, we write $b$ as a Power series in $p$ with a given precison. Is It possible to write $b^k$ as a Power series in $k$, with $k$ an integer ? An example : Let $\gamma_1$ and $\gamma_2$ be the 3-adic unit roots of the quadratic equations $x^2+x+3=0$ and $x^2+2x+3=0$ respectively. Let $k$ be an integer. Let

$$C(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1$$

The problem is to show that

$$v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))>=4\qquad (1).$$k^4))>=4.\qquad (1)$$

I know how to write $\gamma_1$ and $\gamma_2$ as a 3-adic power series with given precision, but have no idea how to work with the exponent. I've tried (1) for k integer between 1 and 100 and the valuation is zero or an integer greater than 3.

Let $b$ be p-adic number, we write $b$ as a Power series in $p$ with a given precison. Is It possible to write $b^k$ as a Power series in $k$, with $k$ an integer ? An example : Let $\gamma_1$ and $\gamma_2$ be the 3-adic unit roots of the quadratic equations $x^2+x+3=0$ and $x^2+2x+3=0$ respectively. Let $k$ be an integer. Let

$$C(k) $$c(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1$$

The problem is to show that

$$v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))>=4.\qquad k^4))\geq 4.\qquad (1)$$

I know how to write $\gamma_1$ and $\gamma_2$ as a 3-adic power series with given precision, but have no idea how to work with the exponent. I've tried (1) for k integer between 1 and 100 and the valuation is zero or an integer greater than 3.

Let $b$ be p-adic number, we write $b$ as a Power series in $p$ with a given precison. Is It possible to write $b^k$ as a Power series in $k$, with $k$ an integer ? An example : Let $\gamma_1$ and $\gamma_2$ be the 3-adic unit roots of the quadratic equations $x^2+x+3=0$ and $x^2+2x+3=0$ respectively. Let $k$ be an integer. Let

$$c(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1$$

The problem is to show that

$$v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))\geq 4.\qquad (1)$$

I know how to write $\gamma_1$ and $\gamma_2$ as a 3-adic power series with given precision, but have no idea how to work with the exponent. I've tried (1) for k integer between 1 and 100 and the valuation is zero or an integer greater than 3.

Let $b$ be p-adic number, we write $b$ as a Power series in $p$ with a given precison. Is It possible to write $b^k$ as a Power series in $k$, with $k$ an integer ? An example : Let $\gamma_1$ and $\gamma_2$ be the 3-adic unit roots of the quadratic equations $x^2+x+3=0$ and $x^2+2x+3=0$ respectively. Let $k$ be an integer. Let

$$c(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1$$

The problem is to show that

$$v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))\geq 4.\qquad (1)$$

I know how to write $\gamma_1$ and $\gamma_2$ as a 3-adic power series with given precision, but have no idea how to work with the exponent. I've tried (1) for k integer between 1 and 100 and the valuation inequality is zero or an integer greater than 3.true only for even numbers. For odd numbers, the left side of (1) is zero.

Let $b$ be p-adic number, we write $b$ as a Power series in $p$ with a given precison. Is It possible to write $b^k$ as a Power series in $k$, with $k$ an integer ? An example : Let $\gamma_1$ and $\gamma_2$ be the 3-adic unit roots of the quadratic equations $x^2+x+3=0$ and $x^2+2x+3=0$ respectively. Let $k$ be an integer. Let

$$c(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1$$

The problem is to show that

$$v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))\geq 4.\qquad (1)$$

I know how to write $\gamma_1$ and $\gamma_2$ as a 3-adic power series with given precision, but have no idea how to work with the exponent. I've tried (1) for k integer between 1 and 100 and the inequality is true only for even numbers. For odd numbers, the left side of (1) is zero.zero.

The inequality (1) is from the article Numerical experiments on families of modular forms by Coleman, Stevens, and Teitelbaum, page 7.