- #1
mathsss2
- 38
- 0
Use:
[tex]2\cdot4\cdot...\cdot(p-1)\equiv(2-p)(4-p)\cdot...\cdot(p-1-p)\equiv(-1)^{\frac{(p-1)}{2}}\cdot1\cdot3\cdot...\cdot(p-2)[/tex] mod [tex]p[/tex]
and
[tex](p-1)!\equiv-1[/tex] mod p [Wilson's Theorem]
to prove
[tex]1^2\cdot3^2\cdot5^2\cdot...\cdot(p-2)^2\equiv(-1)^{\frac{(p-1)}{2}}[/tex] mod [tex]p[/tex]
Relevant equations
Gauss lemma
wilson's theorem [[tex](p-1)!\equiv-1[/tex] mod[tex]p[/tex]]
The attempt at a solution
need assistance
Thanks
[tex]2\cdot4\cdot...\cdot(p-1)\equiv(2-p)(4-p)\cdot...\cdot(p-1-p)\equiv(-1)^{\frac{(p-1)}{2}}\cdot1\cdot3\cdot...\cdot(p-2)[/tex] mod [tex]p[/tex]
and
[tex](p-1)!\equiv-1[/tex] mod p [Wilson's Theorem]
to prove
[tex]1^2\cdot3^2\cdot5^2\cdot...\cdot(p-2)^2\equiv(-1)^{\frac{(p-1)}{2}}[/tex] mod [tex]p[/tex]
Relevant equations
Gauss lemma
wilson's theorem [[tex](p-1)!\equiv-1[/tex] mod[tex]p[/tex]]
The attempt at a solution
need assistance
Thanks