MHB How did we find sqrt(ε_7(2)) ?

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The discussion focuses on the function ε_p that maps integers to p-adic integers, specifically examining ε_7(2) and its square root. The square root of 2 modulo 7 is identified as 3, while for 7^2 (49) it is 10, and for 7^3 (343) it is 108, illustrating the process of finding these roots. The conversation emphasizes the importance of defining square roots in modular arithmetic without using the traditional square root symbol due to its multivalued nature. A correction is noted regarding a typo in the fourth term of the square root sequence. Overall, the thread provides insights into the methodology of calculating square roots in p-adic contexts.
evinda
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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)
 
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evinda said:
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?
 
Last edited:
Bacterius said:
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)
 
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