Pan 2011 integers not form

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Hao Pan, On the integers not of the form p + 2^a + 2^b. Acta Arith. 148 (2011), no. 1, 55-61.

Let N be the set of odd integers not of the form p + 2^a + 2^b and N* the analog with p replaced by a prime power p^alpha. Crocker had shown N is infinite, and Erdos asked whether |N ∩ [1,x]| >> x^epsilon for some epsilon > 0; Granville and Soundararajan noted this follows from unproven assumptions about composite Fermat numbers, and Chen, Feng and Templier obtained conditional lower bounds. Theorem 1.1 proves unconditionally that |N* ∩ [1,x]| >> x exp(-C log x log log log log x / log log log x) for an absolute C > 0, which in particular gives |N* ∩ [1,x]| >> x^{1-epsilon} for every epsilon > 0. The same lower bound holds for N because N* is a subset of N. In the proof, Pan establishes the bound for N and subtracts the O(sqrt(x) log x) integers represented using higher prime powers to obtain the bound for N*. The proof follows Tao's sieve approach: a Brun-Titchmarsh input (Lemma 2.1) bounds the count of n <= x with Wn + beta prime, and a well-chosen modulus W built from many small primes forces most candidate representations to fail. This gives a quantitative lower bound for the exceptional set in Problem 9. It does not establish the positive upper density requested there.

Source: published PDF, Acta Arithmetica 148.1 (2011), pp. 55–61, https://doi.org/10.4064/aa148-1-4. The publisher's record (https://www.impan.pl/get/doi/10.4064/aa148-1-4, read 2026-10-02) labels the download "Free download under CC-BY license", a Creative Commons Attribution license with no version named; the file prints "© Instytut Matematyczny PAN, 2011" on its first page.

Statement check, 2026-09-05. Theorem 1.1 on printed p. 56 (physical page 2) has the factor log x in its exponential, not sqrt(log x). The latter was a transcription error in this digest. The remark on printed p. 60 (physical page 6), stated there without proof, with its parameter K = 1 explicitly includes nonnegative exponents a,b >= 0. Thus that convention is covered by the source; this check does not reconstruct the full proof or certify the present status of Problem 9. The nonrepresentation sets are defined on printed p. 55.

Bears on. #9

Results to transcribe.

• Theorem 1.1: |N* ∩ [1,x]| >> x exp(-C log x * log log log log x / log log log x) for an absolute constant C > 0, hence >> x^{1-epsilon} for all epsilon > 0.
• Lemma 2.1: Brun-Titchmarsh bound: for coprime W, beta the count of n <= x with Wn + beta prime is at most C_1 x/log x times prod_{p|W}(1-1/p)^{-1}.