11.1.15

Life of Phi 2

In Mathematics a single counter-example is detrimental to even the most promising conjecture.

Such was the case, I thought, with Conjecture 2 from Finding Phi where I boldly predicted that the order of 2 mod n where n is the product of 2 distinct primes both greater than 2 was φ(n)/2.

I found a single counter-example: in one instance for n, φ(n)/4 was the smallest integer such that

2φ(n)/4 = 1 mod n

In that instance φ(n)/4 was the order of 2 mod n, which contradicted my conjecture that only φ(n)/2 could be the order of 2 mod n where n = p * q and p & q are two distinct primes both greater than 2. In other words, I thought φ(n)/2 was the smallest integer for which this equation 2φ(n)/2 = 1 mod n is true but it turns out that in some instances 2 has a smaller exponent that divides φ(n)/2.