27.9.19

Goldbach Conjecture

While reading about the Goldbach conjecture recently, I thought of Z35.

A few years ago I conjectured that there are 2 integers a and b such that a2 = 1 mod n and b2 = 1 mod n where a + b = n whenever n is the product of two distinct prime integers greater than 2.

According to the Goldbach Conjecture every even integer greater than 2 can be represented as the sum of two prime integers.

In this respect,  Z35 is very special because:
35 = 5*7
292 = 1 mod 35
and 29+7  = 36 = 1 mod 35

Furthermore,
29+5 = 34 mod 35
342 = 1 mod 35

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