Granville 1990 note sums primes

../

theorem: Granville's theorem that, on the prime k-tuplets conjecture, any positive weights c_1, ..., c_N admit infinite sets A_1, ..., A_N of distinct odd primes with every element of (1/d){c_1 A_1 + ... + c_N A_N} prime, with its two consequences of p. 452.

---

Granville, Andrew, A note on sums of primes. Canad. Math. Bull. 33(4) (1990), 452--454. The copy read for this card is the publisher PDF (three pages). It prints "©Canadian Mathematical Society 1990." at the foot of its first page, and each page carries the line "https://doi.org/10.4153/CMB-1990-073-7 Published online by Cambridge University Press"; the copyright notice is recorded as reserved.

Granville extends to infinite sets the finite constructions of Pomerance, Sárközy and Stewart, conditional on the Hardy--Littlewood prime k-tuplets conjecture. The Theorem (p. 452) states that, given positive integers c_1, ..., c_N, one can construct infinite sets A_1, ..., A_N of distinct odd primes such that every element of (1/d){c_1 A_1 + ... + c_N A_N} is prime, where g = gcd(c_1, ..., c_N) and d = gcd(2g, c_1 + ... + c_N); by the Remark, c_1 A_1 + ... + c_N A_N is the set of sums of any c_1 elements of A_1, any c_2 elements of A_2, and so on. Taking N = 2 with c_1 = 1, c_2 = 2 gives infinite sets A, B with every element of A + B prime, and taking N = 1 with c_1 = 2 gives an infinite set A with (a + a')/2 prime for any a, a' in A (both p. 452). The proof (pp. 452--453) first finds starting primes by a lemma, then adds one prime to each set in turn, chosen by the k-tuplets conjecture so that all the new sums are prime. Erdős problem 431 cites the paper: it asks for infinite sets A and B with A + B equal to the set of primes up to finitely many exceptions. Granville's conditional sets have A + B contained in the primes, which does not answer that question, since A + B need not contain all but finitely many primes.

Source: https://doi.org/10.4153/cmb-1990-073-7.

Bears on. #431: the Theorem's first consequence (p. 452) gives, conditionally on the prime k-tuplets conjecture, infinite sets A, B with every element of A + B prime; the problem asks for A + B to agree with the primes up to finitely many exceptions, which the paper does not address.

Results.

• Theorem (p. 452): on the prime k-tuplets conjecture, infinite sets A_1, ..., A_N of distinct odd primes with every element of (1/d){c_1 A_1 + ... + c_N A_N} prime, with its two unlabelled consequences of p. 452 (A + B prime; (a + a')/2 prime). The lemma of p. 452, used only in the proof, is summarized in that page's proof pointer.

Read status. Claims checked: the Theorem, the Remark, the two consequences and the lemma were read clause by clause against the print, pp. 452--453.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.