Legendre symbol

• May 6th 2008, 03:18 PM
Rakesh
Legendre symbol
Determine the values of

47/37 and

3/43

stating any properties of the legendre symbol used.

is there just one method to solve all of these or are there different methods to solve depending on the numbers.
• May 6th 2008, 07:19 PM
ThePerfectHacker
Quote:

Originally Posted by Rakesh
47/37

$(47/37) = (10/37)$ ---> Congruence
$=(2/37)(5/37)$ ---> Multiplicative

$(2/37) = -1$ because $37\equiv \pm 3(\bmod 8)$.
$(5/37) = (37/5)$ ---> Quadradic Reciprocity
$(37/5) = (2/5)$ ---> Congruence
$(2/5) = 1$ because $5\equiv \pm 1(\bmod 8)$.

Thus, $(10/37) = (2/37)(5/37) = (-1)(1)=-1$.
• May 6th 2008, 10:30 PM
Isomorphism
Quote:

Originally Posted by ThePerfectHacker
$(2/5) = 1$ because $5\equiv \pm 1(\bmod 8)$.

Thus, $(10/37) = (2/37)(5/37) = (-1)(1)=-1$.

Shouldnt the above read like this?

$(2/5) = -1$ because $5\equiv \pm 3(\bmod 8)$.

Thus, $(10/37) = (2/37)(5/37) = (-1)(-1)= 1$.
• May 7th 2008, 12:30 AM
jtsab
Quote:

Originally Posted by Isomorphism
Shouldnt the above read like this?

$(2/5) = -1$ because $5\equiv \pm 3(\bmod 8)$.

Thus, $(10/37) = (2/37)(5/37) = (-1)(-1)= 1$.

That's what I thought, because $7 \equiv -1 \mod8$ and therefore not 5.

Also by Quadratic Reciprocity $\frac {2}{5} =-1$ as 2 is clearly not a square mod 5
• May 7th 2008, 09:36 AM
ThePerfectHacker
Quote:

Originally Posted by Isomorphism
Shouldnt the above read like this?

$(2/5) = -1$ because $5\equiv \pm 3(\bmod 8)$.

Thus, $(10/37) = (2/37)(5/37) = (-1)(-1)= 1$.

Quote:

Originally Posted by jtsab
That's what I thought, because $7 \equiv -1 \mod8$ and therefore not 5.

Also by Quadratic Reciprocity $\frac {2}{5} =-1$ as 2 is clearly not a square mod 5

Yes. It is exactly how Isomorphism said it should be.