Smooth Interpolation, Hölder Continuity, and the Takagi-van der Waerden Function

1 pi − 2 j+1 > 7pr − 4pr > pr . If r is odd, set nk = ∏r1 pi − 2k+1. Then nk ≡ 3(4) (since 32 ≡ 1(4)). But no pi divides nk for 1 ≤ i ≤ r , so the integer nk has some prime factor qk ≡ 3(4) with qk > pr . If j = k, say j > k, the assumption that qk also divides n j leads to the same contradiction as earlier: since nk − n j = 2 j+1 − 2k+1 = 2k+1(2 j−k − 1), we have qk | 2 j−k − 1 and hence qk < pr . Thus, there are at least log2(4r) + 1 distinct primes of the form 4 + 3 in (pr ,∏r1 pi ). If r is even, the same argument applied to r − 1 shows that there are at least log2(4(r − 1)) + 1 distinct primes of the form 4 + 3 in (pr−1, ∏r−1 1 pi ). Since the first of these is pr , there are at least log2(4(r − 1)) distinct primes of the form 4 + 3 in (pr ,∏r−1 1 pi ), a subinterval of (pr ,∏r1 pi ).