For a prime p of the form 4k 1 prove 12 middot 32 middot 52

For a prime p of the form 4k + 1, prove 1^2 middot 3^2 middot 5^2 ... (p - 2)^2 identicalto -1 (mod p). Today is Thursday, what is the day of the week after 1111^1111 days?

Solution

For this we need to find 1111^1111 mod 7

1111=-2 mod 7

So, 1111^1111 mod 7=(-2)^1111 mod 7=(-2)^{1110+1} mod 7

                          = ((-2)^1110)*2=((-2^3)^370)*2 mod 7

                           =((-8)^370)*2 mod 7

                         = ((-1)^370)*2 mod 7

                         =1*2=2 mod 7

So effectively 2 days after Thursday which is Sunday.