_{35}.

According to the Goldbach Conjecture every even integer greater than 2 can be represented as the sum of two prime integers. But, I wondered, how does this conjecture affect odd integers? I also wondered if this conjecture has some application to some of the conjectures I made related to the product of two distinct prime integers greater than 2.

By definition of even and odd integers, if x is an odd integer, then x+1 is an even integer and x-1 is also an even integer.

A few years ago I conjectured that there are 2 non-trivial integers a and b such that a

^{2}= 1 mod n and b

^{2}= 1 mod n where a + b = n whenever n is the product of two distinct prime integers greater than 2. I called a and b "lonely squares". There are also 2 trivial lonely squares, namely 1 and n-1.

Z

_{35}is very special because:

35 = 5*7, which is not an even integer and yet:

29

^{2}= 1 mod 35

and 29+7 = 36 = 1 mod 35

Furthermore,

29+5 = 34 mod 35

34

^{2}= 1 mod 35

Note: 29 and 6 are the only non-trivial "lonely square" in Z

_{35}.

1 is a trivial "lonely square" since 1^2 = 1 for all n where n = p*q where p and q are distinct odd primes.

34 is also a trivial "lonely square" because, as I proved previously, given any n = p*q where p and q are distinct odd primes, then (n-1)^2 = 1 mod n

29 + 34 is clearly not equal to 35, which is okay since 34 is not prime. The interesting part in this case is the fact that there is a direct relation between the odd integer x=35, its surrounding even integers 34 and 36, the prime integers 5 and 7 of which 35 is a product, and the so-called "lonely squares" of Z

_{35}

Another finite field with these interesting properties is Z

_{15}because:

15 = 3*5

11^2 = 1 mod 15

11+ 5 = 16 = 1 mod 15

Furthermore,

11+3 = 14

14^2 = 1 mod 15

Note: 11 and 4 are the only non-trivial "lonely squares" in Z

_{15}. And 14 is a trivial one.

Unfortunately, this is not applicable to all products of primes but it is interesting to see where it pops up and why.

To understand why this happens, one must explore the commonalities between the two examples. Both 35 and 15 have a common factor, namely 5. However, Z

_{55}and Z

_{85}do not exhibit the same properties as Z

_{15}and Z

_{35}. So this is most likely a dead end.

The other thing that both examples have in common is that q = p+2, also known as twin primes. Indeed, it appears that this has something to do with why Z

_{15}and Z

_{35}have such interesting properties.

For example, 11*13 = 143 and the field Z

_{143}exhibits the same interesting properties as Z

_{15}and Z

_{35}since:

143 = 11*13

131^2 = 1 mod 143

131+13 = 144 = 1 mod 143

Furthermore,

131+11 = 142 mod 143

142^2 = 1 mod 143

Is this true for all products of twin primes?