Question

help me with this doubt

Answer 1

given a quadratic congruence that x^2 cong -1 mod p. Where p is an odd prime. Consulting burton's book I have found that (p-1/2)! satisfies the expression.

Now for instance p is taken to be 13 . Hence the solution will be 6!=720. Which means 720 much satisfy the expression. But I am confused that there the congruence of 720 is taken to mod 13 yields 720 cong 5 mod13 and 5 is implied into the solution. Please help me understand this. @ganeshie8

Answer 2

Ref:

Answer 3

heyy

Answer 4

@ganeshie8

Answer 5

still stuck on this ?

Answer 6

yes !!! :(

Answer 7

@ganeshie8

Answer 8

consider the factors in \((p-1)! = (p-1)(p-2)\cdots 3\cdot 2\cdot 1\)

Answer 9

would you agree
\[\large p-1\equiv -1 \pmod{p}\]

?

Answer 10

yes!

Answer 11

thats simply because \(p\equiv 0 \pmod p\)

Answer 12

so \(p-1\equiv 0-1\equiv -1\pmod p\)

Answer 13

I am confused with this fact that 6! is a solution to the equation but why is that 6! was again taken with modulo. I am getting confused with this part basically.

Answer 14

\(x = 6! \) is a solution to the congruence \(x^2\equiv -1\pmod {13}\)

Answer 15

plugin \(x = 6!\) into the conruence and reduec :

\[6!^2 \equiv -1 \pmod{13}\]

Answer 16

\[720^2\equiv -1 \pmod {13}\]

Answer 17

but we know \(720\equiv 5 \pmod{13}\) so we have
\[5^2\equiv -1\pmod{13}\]

Answer 18

\[25\equiv -1\pmod{13}\]

which is true, so yes \(x=6!\) is indeed a solution to the congruence \(x^2\equiv -1\pmod{13}\)

Answer 19

Okay! I know my question is foolish, but why do I need to take the mod of 720 again. :(

Answer 20

I mean 720 itself is the solution. right ???

Answer 21

thats right, 720 itself is the solution
we're just checking whether 720^2 reduces to -1 or not

Answer 22

Also notie, 720 is same as 5 in modulo 13
so essentially \(5\) is a solution to the conruence \(x^2 \equiv -1 \pmod{ 13}\)

Answer 23

oho. sorry for the turmoil... Thank you :)

Answer 24

\[\color{Red}{720}^2\equiv -1 \pmod{13}\]

is same as

\[\color{Red}{5}^2\equiv -1 \pmod{13}\]

Answer 25

np :) it is good to ask questions and make sure every little thing makes sense