Recently a good deal of interest has been taken in the properties of numbers which can be expressed and proved without going beyond the number 1 and the Four Rules, in other words by using only rational numbers. Professor T. S. Broderick has established various theorems about the number and distribution of primes, using this “sub-language”: (Journal of the London Mathematical Society, Vol. 14, pp. 303–310; Proceedings of the Royal Irish Academy, 1940, pp. 17–24).