Question

Number theory:

"Use Wilson's theorem
(p-1)! congruent to -1(mod p)
to show that
(p-3)1 congruent to ((p-1)/2) mod p"

Thanks :)

Answer 1

the only thing that comes to mind at the moment is tom hanks yelling at a volleyball :/

Answer 2

i spose Wilsons thrm is: (p-1)! = -1 (mod p)

Answer 3

yup

Answer 4

-3 = -1-2

Answer 5

([p-2] -1) ! = -1 (mod [p-2])

but that doesnt seem right to me .. just yet

Answer 6

(p-1)! = (p-1)(p-2)(p-3)!

Answer 7

that should prolly be used to start out with

Answer 8

multiply both sides by p-1 p-2, or i might be working it backwards ..

Answer 9

(p-1)! = -1(mod p)

(p-1)(p-2)(p-3)! = -1 (mod p)

need to develop some inverses

Answer 10

1 2 3 4 5

1 1 2 3 4 0
2 2 4 1 3 0
3 3 1 4 2 0
4 4 3 2 1 0
5 0 0 0 0 0

seems to be a good representation of a multiplication table in mod 5

Answer 11

4! = -1 (m5)

432! = -1 (m5)

4432! = -14 (m5)

232! = -142 (m5)

2! = -8 (m5)

-8+5+5 = 2

2! = 2 (m5) = (4-1)/2

Answer 12

(p-1)(p-1)/p


p -2
-----------
p | p^2-2p+1
(p^2)
------------
-2p+ 1
(-2p)
--------
1 <-- remainder of 1

(p-1)(p-1) = 1 (mod p)

Answer 13

(p-2)(p-k)/p


p -(2+k)
------------------
p | p^2 -(2+k)p +2k
(p^2)
-----------------
-(2+k)p +2k
[(-2+k)p]
--------------
2k (mod p) for some k

not too sure if that can help us out .....

Answer 14

(p-1)(p-2)(p-3)! = -1 (mod p) ; lets use -1 = (p-1)

(p-1)(p-2)(p-3)! = (p-1) (mod p)

(p-1)(p-1)(p-2)(p-3)! = (p-1)(p-1) (mod p)

(p-2)(p-3)! = 1 (mod p)

(p-k)(p-2)(p-3)! = (p-k)1 (mod p)

(p-3)! = (p-k) (mod p)

(p-3)! = 2k (mod p)

Answer 15

thats wrong a little bit i think ... but is it giving you any ideas?

Answer 16

(p-k) = k (mod p)

Answer 17

if we can show that k = (p-1)/2 youd be set

Answer 18

all thats really left to show .. is that \[\Large \frac{(p-1)}{2}(p-2)\equiv1~(mod ~p)\]
(p-1)(p-2) = p^2 -3p +2


p - 3
---------------
p | p^2 -3p +2
(p^2)
------
-3p
(-3p)
------
2; and 1/2 * 2 = 1 for a remainder

Answer 19

I get it! Thankyou so much!
I think it was the factorial sign that was throwing me off!

Answer 20

it was a little something else throwing me at first :) but i figured it out