**Definition:**Given two distinct odd primes p and q and an integer

**< p*q such that GCD(**

*a***, p*q) = 1 and**

*a*

*a*^{2}= 1 mod p*q then

**a**is a lonely square mod p*q.

Clearly,

**= 1 is the most trivial example of a lonely square.**

*a*In Types of Exponents I showed that (p*q - 1)

^{2}= 1 mod pq therefore

**= (p*q - 1) is a lonely square mod p*q.**

*a***Claim:**If 2 has order k mod p*q where p and q are distinct odd primes and k is an even integer then (2k/2) is a lonely square.

**Proof:**Since k is the order of 2

_{ }then 2k= 1 mod p*q

Since k is an even integer then k/2 is also an integer and so:

(2k/2)

^{2}mod p*q = (2(k/2)*2) mod p*q = 2k mod p*q = 1 mod p*q

**Example:**Let p = 3, q = 7.

To find the order of 2 mod 21 using my favourite algorithm I first list all even integers < 21 followed by all odd integers < 21, (which is the second row of the multiplication table of Z

_{21})

Then starting from index 1 I look up the value at that index, which is 2, then I look up the value at index 2, which is 4, then I look up the value at index 4, which is 8, then I look up the value at index 8, which is 16, then I look up the value at index 16, which is 11, then I look up the value at 11, which is 1, and this completes the cycle [a

_{1}

^{0},a

_{1}

^{1}, a

_{1}

^{2}, a

_{1}

^{3},..., a

_{1}

^{k}], which is equivalent to [2

^{0},2

^{1}, 2

^{2}, 2

^{3},..., 2

^{6}] or [1, 2, 4, 8, 16, 11, 1] (which is the unique portion of the second row of the exponentiation table of Z

_{21})

Note: I first discussed the basis for this algorithm here.

So k = 6, which is even and the value at k/2 is 8. It is clear to see that 8

^{2}= 1 mod 21

**Claim:**If 2k/2 = (p*q - 1) mod p*q where k is the order of 2 mod p*q and it is even and p, q > 2 then there is at least one other small prime b of order r where r is even such that b does not belong to the set [2

^{0},2

^{1}, 2

^{2}, 2

^{3},..., 2

^{k}] but b

^{r/2}mod p*q is a lonely square.

**Claim:**If p, q > 2 then there are at least 4 lonely squares.

**Claim:**If d

^{s}mod p*q is a lonely square then there is another lonely square c where:

c = p*q - d

^{s}