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On integers of the form \(2^k+p\) and some related problems. (English) Zbl 0041.36808

Summa Brasil. Math. 2, 113-123 (1950).
If \(n\) is a positive integer, let \(f(n)\) denote the number of solutions of the equation \(2^k+p=n\), \(k\) being a non-negative integer and \(p\) a prime number. The author proves that:
(I) there is a positive constant \(c_0\) such that \(f(n) > c_0 \log\log n\) for infinitely many positive integers \(n\),
(II) if \(r\) is a fixed positive integer, there exists a positive number \(c(r)\) such that \(\sum_{n=1}^x f^r(n) < c(r)x\),
(III) if \(n\equiv 7629217 \pmod{11184810}\); \(f(n)=0\).
(II) contains a theorem of N. P. Romanoff [Math. Ann. 109, 668–678 (1934; Zbl 0009.00801; JFM 60.0131.03)] to the effect that the positive integers expressible in the form \(2^k+p\) have positive density. The author generalizes Romanoff’s theorem by proving, that if \(a_1 < a_2 < \cdots\) is an infinite sequence of positive integers such that \(a_k \mid a_{k+1}\) for each \(k\), then the positive integers expressible in the form \(p+a_k\) have positive density if and only if there exist positive numbers \(c_1\) and \(c_2\) such that \(\frac{\log a_k}{k} < c_1\) and \(\sum_{d/a_k} \frac{1}{d} < c_2\) for every \(k\).
Reviewer’s remark: The proof of (I) uses a result of A. Page [Proc. Lond. Math. Soc. (2) 39, 116–141 (1935; Zbl 0011.14905)] on the number of prime numbers in an arithmetic progression with relatively large difference. In applying this result, the author forgets to take account of a possible exceptional real primitive residue character which occurs in Page’s work. This difficulty can be easily overcome in much the same manner as an analogous difficulty was handled in a joint paper of the author, the reviewer, and S. Chowla [Publ. Math. 1, 165–182 (1950; Zbl 0036.30702), see p. 170 of the work].
Reviewer: Paul T. Bateman

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes