Takagi defined a class field to be one where equality holds in the second inequality.

But the equality does not hold for greedoids in general.

If the equality does hold for many values of a, then we can say that n is probably prime, or a pseudoprime.

The last equality above holds because there are different pairs .

If is continuous, then equality holds in the former.

At any other epoch, when the equality did not hold, there would be no intelligent life around to notice the discrepancy.

If we get 1 then the second equality holds and it is done.

This equality would have held for any chosen point .

If the equality does not hold for a value of a, then p is composite.

If the equality does hold for many values of a, then we can say that p is probable prime.