Note on Some Additive Properties of Integers

Let A=2 .3 . . . p r , the product of consecutive primes, be sufficiently large . By elementary method we prove that the number of solutions of the congruence p2-q2 =-0 (mod A) with 0 < q < p < A is greater than 1 A 1 + 41og log A . But the integers of the form p 2-q 2 with 0 < q< p < A lie all between 0 and A', hence there exists a multiple of 4 say n(<,4") such that the number of solutions of the equation n=p 2-q2 is greater than 1 1 A4 log log A > n8 log log n