# Solutions to x^2==22(mod103) in Z103

• clueles
In summary, to find the number of solutions in z103 for the equation x^2==22(mod103), you need to determine the value of the Legendre symbol \left(\frac{22}{103}\right). This can be done using the 3-part Law of Quadratic Reciprocity, which involves calculating the Legendre symbol for -1, 2, and the quotient of p and a.
clueles
find the number of solutions in z103

x^2==22(mod103)

no i haven't

Legendre symbol?

The Legendre symbol $$\left(\frac{a}{p}\right)$$ is defined as 1 if $$x^2\equiv a\pmod p$$ has solutions, and -1 otherwise. (It's undefined or 0 if $$p\mid a$$.)

Thus for $$x^2\equiv22\pmod{103}$$ you're trying to decide the value of the Legendre symbol $$\left(\frac{22}{103}\right)$$.

Here's the 3-part Law of Quadratic Reciprocity:

$$\left(\frac{-1}{p}\right)=(-1)^{\frac{p-1}{2}}$$

$$\left(\frac{2}{p}\right)=(-1)^{\frac{p^2-1}{8}}$$

$$\left(\frac{a}{p}\right)=(-1)^{\frac{(p-1)(a-1)}{4}}\left(\frac{p}{a}\right)$$

(If you're using a definition that doesn't include 0, you can move the two Legendre symbols to the same side for aesthetics.)

## 1. What is the definition of a "solution" in this context?

A solution in this context refers to a number that, when squared, is equivalent to 22 modulo 103 in the set of integers modulo 103 (also known as Z103).

## 2. How many solutions are there to this equation?

In the set of integers modulo 103, there are either two solutions or no solutions to this equation. This is because the set of integers modulo 103 has a total of 103 elements, and for any given number, there can only be two possible square roots modulo 103.

## 3. How can I find the solutions to this equation?

There are a few methods for finding solutions to this equation in Z103. One approach is to use trial and error, plugging in different numbers and checking if their square is equivalent to 22 modulo 103. Another method is to use modular arithmetic and algebraic manipulation to simplify the equation and find the solutions.

## 4. Can there be more than one solution for a given number in Z103?

No, there can only be a maximum of two solutions for a given number in Z103. This is because of the nature of modular arithmetic - for any given number, there can only be two possible square roots modulo 103.

## 5. Are there other ways to represent the solutions to this equation?

Yes, instead of using the set of integers modulo 103, the solutions to this equation can also be represented using congruence notation. For example, if x is a solution, then x ≡ ± √22 (mod 103). This notation indicates that x is equivalent to either the positive or negative square root of 22 modulo 103.

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