Chen 2023 conjecture erdos p 2 k

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theorem_1_1: Chen's theorem that for every set S of asymptotic density zero the union of S with the positive odd integers not of the form p + 2^k (p prime, k >= 1) is not a union of finitely many infinite arithmetic progressions and a set of asymptotic density zero; Corollary 1.2 is the case S empty.

theorem_1_15: Chen's theorem that the union of all infinite arithmetic progressions contained in the non-representable odd integers equals the union of those whose common difference is a power of 2 times a squarefree odd integer, and, if there are infinitely many Mersenne primes, the union of those with squarefree common difference; Corollary 1.16 is the unconditional dichotomy with Problem 1.11.

theorem_1_3: Chen's theorem that over infinite progressions mh + a whose part outside the non-representable odd integers has density zero, min m = 11184810 and min omega(m) = 7, with omega(m) = 7 only for m = 11184810; Corollary 1.4 gives the same for progressions contained in that set.

theorem_1_5: Chen's determination of the residues a for which the progression 11184810h + a is a longest quasi-non-representable infinite arithmetic progression, a list (1.1) of 48 odd residues, with Corollary 1.6 that 11184810h + b is contained in the non-representable odd integers exactly when b >= 0 and b is congruent to a residue on that list.

theorem_1_9: Chen's criterion that an element a of the non-representable odd integers lies in an infinite arithmetic progression of non-representable odd integers if and only if some integer m > 1 has gcd(a - 2^k, m) > 1 for every positive integer k, with the equivalent Corollary 1.10 in terms of least prime divisors.

theorem_3_1: Chen's statement that Erdős's Conjecture A fails, the non-representable odd integers not being one infinite arithmetic progression plus a set of density zero, with two proofs independent of Theorem 1.1: one from two explicit progressions modulo 11184810 inside the set, one from Theorems 1.3 and 1.5.

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Yong-Gao Chen, A conjecture of Erdős on p+2^k. arXiv preprint (2023). arXiv:2312.04120, doi:10.48550/arXiv.2312.04120. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2312.04120), every other right reserved. The copy read for this card is arXiv:2312.04120v3 (18 February 2024).

Chen studies the set U of positive odd integers that cannot be written as the sum of a prime and a power of two, p + 2^k with k a positive integer. Erdos conjectured (Conjecture A, p. 2) that U is the union of one infinite arithmetic progression of odd integers and a set of asymptotic density zero; the paper identifies this as Problem 16 of Bloom's list. Theorem 1.1 (p. 2) refutes it in a stronger form: for every set S of asymptotic density zero, the union of U and S is not a union of finitely many infinite arithmetic progressions and a set of density zero; Corollary 1.2 is the case S empty. Section 3 gives a second proof that Conjecture A is false (Theorem 3.1, p. 13) from two explicit progressions modulo 11184810 contained in U (Lemmas 3.3 and 3.4). Theorem 1.3 (p. 2) shows that among infinite progressions mh+a that lie in U up to a set of density zero, the least modulus is m = 11184810 and the least number of distinct prime factors of m is 7, attained only at that modulus; Corollary 1.4 gives the same for progressions contained in U. Theorem 1.5 (p. 3) lists the 48 residues a for which 11184810h+a is a longest quasi-non-representable progression (a > 0, in U up to a set of density zero, and a proper subset of no other such progression), and Corollary 1.6 shows that the progression 11184810h+b, h >= 0, is contained in U exactly when b >= 0 and b is congruent to a residue on that list; Section 4 uses Theorems 1.3 and 1.5 for a third disproof of Conjecture A (p. 25). Theorem 1.9 (p. 4) characterizes the elements a of U that lie in some infinite progression contained in U: those for which some integer m > 1 has gcd(a - 2^k, m) > 1 for every positive integer k. Theorem 1.15 (p. 6) shows that these progressions may be taken with common difference a power of 2 times a squarefree odd integer, and with squarefree common difference if there are infinitely many Mersenne primes. The introduction also poses Problems 1.7, 1.8 and 1.11-1.13 and Conjecture 1.14: the set of positive integers a for which no integer m > 1 satisfies gcd(a - 2^k, m) > 1 for every positive integer k has positive lower asymptotic density.

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages of arXiv v3; no proof is checked step by step.

Source: https://arxiv.org/abs/2312.04120.

Bears on.

• #16: Corollary 1.2 and Theorem 3.1 answer the problem's question no, with k >= 1 as the paper fixes it; Theorem 1.1 rules out finitely many progressions plus a density-zero set, even after adding any density-zero set.
• #236: context only. The paper concerns the integers with no representation n = p + 2^k and proves nothing about the size of the number f(n) of representations that the problem asks about; it uses, as a cited tool, the bound sum_{n <= x} r(n)^2 << x for the number r(n) of representations with k >= 1 (its (2.11), p. 9), which differs from the problem's f(n), counted with k >= 0, by at most 1.

Results.

• Theorem 1.1 (p. 2): For every density-zero S, U together with S is not finitely many infinite progressions plus a density-zero set; Corollary 1.2 takes S empty.
• Theorem 1.3 (p. 2): Over progressions lying in U up to density zero, min m = 11184810 and min omega(m) = 7, with omega(m) = 7 only at m = 11184810; Corollary 1.4 for progressions contained in U.
• Theorem 1.5 (p. 3): The 48 residues of the longest quasi-non-representable progressions modulo 11184810, with Corollary 1.6.
• Theorem 1.9 (p. 4): An element a of U lies in a progression contained in U if and only if some m > 1 has gcd(a - 2^k, m) > 1 for all k >= 1; Corollary 1.10.
• Theorem 1.15 (p. 6): The progressions contained in U may be taken with modulus a power of 2 times a squarefree odd integer, and squarefree if there are infinitely many Mersenne primes; Corollary 1.16.
• Theorem 3.1 (p. 13): Conjecture A is false, proved from Lemmas 3.2-3.4 and again from Theorems 1.3 and 1.5.

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