On Some Empirical Theorems of Scherk

Let

   P1, P2 … Px

be the first natural primes in order so that p₁ = 2, p₂ = 3, etc

Then Scherk's theorems are:

THEOREM I. If n is even, p_n can be obtained by the addition and subtraction of all the smaller primes and unity, each taken once, eg.,

p8=- 19 = 1 + 2 + 3 + 5 - 7 + 11 - 13 + 17.