Arithmetic Tales by Olivier Bordellès

Author:Olivier Bordellès
Language: eng
Format: epub, pdf
Publisher: Springer London, London

We deduce at once the following result.

Corollary 5.33

If Nδ⩽c 0 and 4⩽N⩽x 1/2, then

Furthermore, if x 2/5⩽N⩽x 1/2, then

Note that this result is equivalent to Proposition 5.24 used with k=2 except with the condition Nδ≪1 instead of (5.25).

5.4.3 The Method of Filaseta and Trifonov—Higher Divided Differences

By Corollary 5.33, it is now sufficient to assume

(5.38)

and we set

By (5.38), we have R<A⩽N 1/2. We take up again the subset T of Lemma 5.32 in which any two consecutive elements differ by >R. As in Lemma 5.7, we will use the following subsets of T. For a∈ ]R,A]∩ℤ, we define

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