# Motivation of Jacobi symbol

Mathematics Asked by athos on December 9, 2020

I fancy the motivation of introducing Legendre symbol is to simplify the quadratic residue check.

But, for Jacobi symbol, even when its value is 1, it doesn’t necessary means that its Legendre Symbol is 1: $$left(frac 2 {15}right)_{Jacobi} = left(frac 2 3right)left(frac 2 5right) =-1*-1 = 1$$
, and 2 is not quadratic residue of 15.

So what is the motivation of introducing Jacobi symbol?

Wiki introduced its usage in Primality_testing , "This is the basis for the probabilistic Solovay–Strassen primality test and refinements such as the Baillie-PSW primality test and the Miller–Rabin primality test." Are there other usages?

## Related Questions

### Invariant $SU(3)$ subgroup for ${bf 8}$ in ${bf 3}^* otimes {bf 3} ={bf 1} oplus {bf 8}$

1  Asked on December 3, 2021 by annie-marie-cur

### Proof that a continuous function with continuous right derivatives is differentiable.

1  Asked on December 3, 2021

### Finding the volume when a parabola is rotated about the line $y = 4$.

1  Asked on December 3, 2021

### Differential equation, modulus signs in solution?

2  Asked on December 3, 2021 by refnom95

### Consider the sequence where $a_1>0$, $ka_n>a_{n+1}$ and $0<k<1$. Can we say it converges?

1  Asked on December 3, 2021 by oek-cafu

### In a Reflexive banach space, given a closed convex set $C$ and some point $y$, there is a point in $C$, of minimal distance to $y$

2  Asked on December 3, 2021

### Ball / Urn question with a twist

1  Asked on December 3, 2021 by user109387

### How to prove $phi'(t)1_{Omega_t}(w)$ is measurable?

1  Asked on December 3, 2021 by czzzzzzz

### Using characteristic functions to determine distribution of sum of independent normal random variables.

0  Asked on December 3, 2021 by jkeg

### Radioactivity formula using differential equations?

3  Asked on December 3, 2021 by mitali-mittal

### What is the algebraic interpretation of a contracted product?

0  Asked on December 3, 2021 by james-steele

### $f:[0,1]rightarrow[0,1]$, measurable, and $int_{[0,1]}f(x)dx=yimplies m{x:f(x)>frac{y}{2}}geqfrac{y}{2}$.

1  Asked on December 3, 2021

### Evaluating an integral with a division by $0$ issue

3  Asked on December 1, 2021

### Evaluating $sumlimits_{i=lceil frac{n}{2}rceil}^inftybinom{2i}{n}frac{1}{2^i}$

1  Asked on December 1, 2021 by maxim-enis

### Lie group theory’s connection to fractional calculus?

0  Asked on December 1, 2021

### Solution to autonomous differential equation with locally lipscitz function

1  Asked on December 1, 2021

### Let $A,Bin M_n (mathbb R), lambdain sigma(B), alpha in sigma (B)$ and  be inner product show $=lambda||x||^2+alpha$

0  Asked on December 1, 2021

### Is an automorphism a function or a group?

2  Asked on December 1, 2021 by josh-charleston

### Fundamental Theorem of Projective Geometry for Finite Fields

0  Asked on December 1, 2021 by am2000

### Prove a metric space is totally bounded

2  Asked on December 1, 2021