These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integer formula_1 larger than 4, for formula_2 a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version (without of course being a counterexample to the original version) of the third conjecture. The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer formula_3, could not possibly rule out the existence of such a specific counterexample formula_4). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement.