How did we find sqrt(ε_7(2)) ?

[evinda]

Hi! (Smile)

Let the function
$$\epsilon_p: \mathbb{Z} \rightarrow \mathbb{Z}_p$$
$$x \mapsto (\overline{x_n} )_{n \in \mathbb{N}_0}$$

where $\overline{x_n} \equiv x \pmod {p^n}$

There are some examples:

$$\epsilon_7 (173)=(173,173,173, \dots, 173, \dots)=(5,26,173,173, \dots)$$

$$\epsilon_7(-1)=(-1,-1,-1, \dots)=(6,48,342,2400, \dots)$$

$$\epsilon_7(2)=(2,2, \dots, 2, \dots)$$

$$\sqrt{\epsilon_7(2)}=(3,10,108,2016, \dots)$$

Could you explain me how we found $\sqrt{\epsilon_7(2)}$ ? (Thinking)

 

[Nono713]

Hi! (Smile)

Let the function
$$\epsilon_p: \mathbb{Z} \rightarrow \mathbb{Z}_p$$
$$x \mapsto (\overline{x_n} )_{n \in \mathbb{N}_0}$$

where $\overline{x_n} \equiv x \pmod {p^n}$

There are some examples:

$$\epsilon_7 (173)=(173,173,173, \dots, 173, \dots)=(5,26,173,173, \dots)$$

$$\epsilon_7(-1)=(-1,-1,-1, \dots)=(6,48,342,2400, \dots)$$

$$\epsilon_7(2)=(2,2, \dots, 2, \dots)$$

$$\sqrt{\epsilon_7(2)}=(3,10,108,2016, \dots)$$

Could you explain me how we found $\sqrt{\epsilon_7(2)}$ ? (Thinking)

A square root of two modulo $7$ is $3$, since $3^2 \equiv 9 \equiv 2 \pmod{7}$. A square root of two modulo $7^2 = 49$ is $10$ since $10^2 \equiv 100 \equiv 2 \pmod{49}$. A square root of two modulo $7^3 = 343$ is $108$ since $108^2 \equiv 11664 \equiv 2 \pmod{343}$. And so on. You can still define square roots modulo $n$, though I would recommend not using the $\sqrt{}$ symbol as it is multivalued and generally a nuisance, just stick with the definition: an integer $x$ is a square root of $a$ modulo $n$ if and only if $x^2 \equiv a \pmod{n}$.

EDIT: I suppose the fourth term should be $2166$, not $2016$. Typo?

 

[evinda]

A square root of two modulo $7$ is $3$, since $3^2 \equiv 9 \equiv 2 \pmod{7}$. A square root of two modulo $7^2 = 49$ is $10$ since $10^2 \equiv 100 \equiv 2 \pmod{49}$. A square root of two modulo $7^3 = 343$ is $108$ since $108^2 \equiv 11664 \equiv 2 \pmod{343}$. And so on. You can still define square roots modulo $n$, though I would recommend not using the $\sqrt{}$ symbol as it is multivalued and generally a nuisance, just stick with the definition: an integer $x$ is a square root of $a$ modulo $n$ if and only if $x^2 \equiv a \pmod{n}$.

EDIT: I suppose the fourth term should be $2166$, not $2016$. Typo?

Yes, that was a typo..

I understand, thank you very much! (Smile)