# Talk:Jacobi symbol

## Promote to Task[edit]

This task is well defined and has many implementation. Any objection to promoting this draft task to a task?

--DavidFashion (talk) 22:55, 5 February 2020 (UTC)

## Lead rewrite[edit]

I rewrote the lead to use LaTeX, but Mathoid wasn't working when I tried to preview it, so I'm dumping it here so that I don't lose it. Disregard the errors if it doesn't re-render by the time you see this. -happy5214 06:22, 6 February 2020 (UTC)

The **Jacobi symbol** is a multiplicative function that generalizes the Legendre symbol. Specifically, the Jacobi symbol **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("<p>There was a problem during the HTTP request: 404 Not Found
</p>") from server "http://mathoid.testme.wmflabs.org":): \left(\frac{a}{n}\right)**

equals the product of the Legendre symbolsFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("<p>There was a problem during the HTTP request: 404 Not Found

**</p>") from server "http://mathoid.testme.wmflabs.org":): \left(\frac{a}{p_i}\right)^{k_i}**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("<p>There was a problem during the HTTP request: 404 Not Found
</p>") from server "http://mathoid.testme.wmflabs.org":): n = p_1^{k_1}p_2^{k_2}\cdots p_i^{k_i}**

is the prime factorization of and the Legendre symbolFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("<p>There was a problem during the HTTP request: 404 Not Found

**</p>") from server "http://mathoid.testme.wmflabs.org":): \left(\frac{a}{p}\right)**

denotes the value ofFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("<p>There was a problem during the HTTP request: 404 Not Found

**</p>") from server "http://mathoid.testme.wmflabs.org":): a^{(p-1)/2} \pmod{p} = \begin{cases} 1 & \text{if a is a square} \pmod{p} \\ -1 & \text{if a is not a square} \pmod{p} \\ 0 & \text{if } a \equiv 0 \pmod{p} \end{cases}**

If n is prime, then the Jacobi symbol **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("<p>There was a problem during the HTTP request: 404 Not Found
</p>") from server "http://mathoid.testme.wmflabs.org":): \left(\frac{a}{n}\right)**

equals the Legendre symbol

**</p>") from server "http://mathoid.testme.wmflabs.org":): \left(\frac{a}{n}\right)**
.

- Task

Calculate the Jacobi symbol **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("<p>There was a problem during the HTTP request: 404 Not Found
</p>") from server "http://mathoid.testme.wmflabs.org":): \left(\frac{a}{n}\right)**
.