The skewy revisited

A few years ago I boldly introduced a new term called "skewy", which I defined as the smallest square root of (n+1/2)² mod n where n is the product of two distinct odd primes.

The skewy has an interesting relation to the Collatz conjecture and to the prime factorization of n.

Here are a few claims to consider:

Claim #1:   4*((n+1)/2)² = 1 mod n

Claim #2:   The row at 4 of the exponentiation table of Z_n consists of all squares co-prime with n.

Claim #4:   If s is the skewy, then 4*(s³ mod n) mod n = s.

Claim #5:   4*(((n+1)/2)³ mod n) mod n = (n+1)/2.