Ten new cases of a conjecture of Chen and Xu on primes plus sparse power sums

Jeff Pickhardt

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Ten new cases of a conjecture of Chen and Xu on primes plus sparse power sums

Abstract

Let (F_n) be the Fibonacci sequence. Chen and Xu conjectured that if r_1,...,r_t are positive integers with r−1 1 +···+ r−1 t = 1, then there is an infinite arithmetic progression of positive integers none of which is representable as p+ X_k^r1 +···+ X_k^rt with p prime and k_i ∈ N_0, both 1 t for X= F and for Xm = 2m. Only the cases t = 1, r_1 = 1 and t= 2, (r_1, r_2) = (2, 2) were known. We establish ten further cases, with t= 3,4,5,6, by exhibiting explicit progressions. The method rests on the observation that the problem is finite: if every prime in the modulus has period dividing Λ, then Xkr mod q depends only on kr mod Λ, so a covering of the finite set of residue states is a proof. We also introduce a lifting step which finishes a covering that a search leaves a few states short, and use it to settle the Fibonacci case (2, 4, 4). We believe the methods in this paper can be used to find more cases.