MATHEMATICS

A. F. LAVRIK

ON THE REPRESENTATION OF NUMBERS IN THE FORM OF THE SUM OF A PRIME NUMBER AND A POWER OF A GIVEN INTEGER

(Presented by Academician I. M. Vinogradov on 7 III 1957)

§ 1. The present paper is devoted to the study of the question of the number of representations of natural numbers \(n\) in the form

\[ n=p+a^i, \]

where \(p\) is a prime number; \(a \geqslant 2\) is a given integer; \(i \geqslant 0\) is an integer.

Theorem 1. In the interval \((0,x)\) there are more than \(\dfrac{\alpha x}{\lg a}\) numbers representable in one and only one way in the form of the sum of a prime number and a power of the given integer \(a>a_0\), where \(\alpha\) is an absolute positive constant.

Without a restriction on the number \(a\), Theorem 2 is valid.

Theorem 2. There exists a constant number \(k\), independent of \(x\) and \(a\), such that the number of numbers \(n \leqslant x\) for which the equation \(n=p+a^i\), where \(p\) is prime, \(a \geqslant 2\) is a given integer, \(i \geqslant 0\) is an integer, has \(1,2,\ldots,k\) solutions, will be more than \(\dfrac{\gamma x}{\lg a}\), where \(\gamma>0\) is a constant.

Theorems 1 and 2 refine known results of N. P. Romanov \((^1)\), E. Landau \((^3)\), and are obtained from a more general theorem.

Theorem 3. Let \(\psi(n,x)\) be the number of solutions of the equation

\[ n=p+a^i, \]

where \(p \leqslant x\) is a prime number, \(a \geqslant 2\) is a given integer, \(i \geqslant 0\) is an integer, \(a^i \leqslant x\). Further, let \(F_m(x)\) be the number of numbers \(n \leqslant 2x\) for which \(\psi(n,x)=m\), \(m>0\) an integer; let \(k\) be any odd positive number. Put also

\[ \Phi_k(x)=F_1(x)+\cdots+F_k(x). \]

Then

\[ \Phi_k(x) \geqslant \frac{4kx}{(k+1)^2 \lg a} \left(1-\frac{c_1 \lg^3 \lg 2a}{k \lg a}-\frac{c_2}{\lg x}\right), \]

where \(c_1,c_2\) are positive absolute constants.

We outline the proof of Theorem 3. We have

\[ \sum_{m=1}^{k} mF_m(x) = \sum_{n=1}^{2x}\psi(n,x) - \sum_{\substack{n=1\\ \psi(n,x)>k}}^{2x}\psi(n,x). \tag{1} \]

Next, we note that the subtracted term in equality (1) does not exceed

\[ \sum_{n=1}^{2x}\psi(n,x)[\psi(n,x)-k] + \sum_{m=1}^{k-1} m(k-m)F_m(x). \tag{2} \]

From (1) and (2) it follows that

\[ \max_{1\le m<k}\{m(k-m+1)\}\Phi_k(x)\ge \sum_{n=1}^{2x}\psi(n,x)[k+1-\psi(n,x)]. \tag{3} \]

Now we shall use the identity of N. P. Romanov \({}^{1}\)

\[ \sum_{n=1}^{2x}\psi^2(n,x)= \sum_{n=1}^{2x}\psi(n,x)+2\sum_{n=1}^{x}A_1(n,x)A_2(n,x), \tag{4} \]

where \(A_1(n,x)\) and \(A_2(n,x)\) are, respectively, the numbers of solutions of the equations:

\[ p_i-p_j=n,\qquad p_i,p_j\le x\text{ — prime numbers;} \]

\[ a^u-a^t=n,\qquad a^u,a^t\le x,\qquad u,t\ge0\text{ — integers.} \]

From (3) and (4) we obtain

\[ \Phi_k(x)\ge \frac{4k}{(k+1)^2}\sum_{n=1}^{2x}\psi(n,x) - \frac{8}{(k+1)^2}\sum_{n=1}^{x}A_1(n,x)A_2(n,x). \tag{5} \]

It is not difficult to see that

\[ \sum_{n=1}^{2x}\psi(n,x)=\pi(x)N(x), \tag{6} \]

where \(\pi(x)\) is the number of primes \(\le x\); \(N(x)\) is the number of numbers \(a^i\le x\).

On the basis of the investigations of Viggo Brun, L. G. Schnirelmann \({}^{2}\), N. P. Romanov \({}^{1}\), and E. Landau,

\[ \sum_{n=1}^{x}A_1(n,x)A_2(n,x)<cx\frac{\lg^3\lg 2a}{\lg^2 a}, \tag{7} \]

where \(c>0\) is an absolute constant.

Combining the estimates (5)—(7), we obtain Theorem 3.

§ 2. The following two propositions are a supplement to Theorems 1 and 2.

Theorem 4. There exists an infinite set of numbers \(n\) for which (in the notation of Theorem 3), as \(x\to\infty\), we have

\[ \psi(n,x)>\delta\lg\lg n, \]

where \(\delta>0\) is some constant.

Theorem 5. If \(\gamma(x)\) is any positive function, increasing without bound as \(x\to\infty\), and \(M(x)\) is the number of numbers \(n\le x\) for which

\[ \psi(n,x)>\gamma(x), \]

then

\[ M(x)=o(x). \]

Tashkent State Pedagogical Institute
named after Nizami

Received
4 III 1957

CITED LITERATURE

\({}^{1}\) N. P. Romanov, Uspekhi Mat. Nauk, 7, 47 (1940).
\({}^{2}\) L. G. Schnirelmann, Uspekhi Mat. Nauk, 7, 7 (1940).
\({}^{3}\) E. Landau, Acta Arithmetica, 1, 43 (1935).
\({}^{4}\) K. Prachar, J. London Math. Soc., 29, No. 3 (1954).