MATHEMATICS

A. F. LAVRIK

ADDITION OF A PRIME NUMBER TO A PRIME POWER OF A GIVEN PRIME

(Presented by Academician I. M. Vinogradov on 19 XII 1957)

§ 1. In the present paper we consider certain additive problems of binary type with prime numbers. In particular, we solve the question of the number of integers \(n\), not exceeding a given bound, for which the equation

\[ n=p_1+p_2^{p_3} \tag{1} \]

is soluble in primes \(p_1, p_2, p_3\), where \(p_2\) is fixed.

Theorem 1. “Almost” all numbers of the sequence \(p_1+p_2^{p_3}\), where \(p_1, p_2\) independently run through prime numbers and \(p_2\) is a fixed prime, are distinct.

In other words, the number of integers representable in the form (1) in more than one way has a smaller order of growth in comparison with the number of integers representable in the form (1), but in only one way.

A more precise fact is supplied by the following theorem.

Theorem 2. Let \(Q(p_2,N)\) be the number of all integers \(n\leqslant 2N\) representable in the form (1) with \(p_1\leqslant N\), \(p_2^{p_3}\leqslant N\); let \(F(p_2,N)\) be the number of those among them which are representable in only one way. The symbol \(\sim\) denotes the sign of asymptotic equality.

Then we have:

\[ Q(p_2,N)=\frac{N}{\ln p_2\cdot \ln\ln N} +O\left(\frac{N\ln\ln\ln N}{\ln p_2\cdot \ln^2\ln N}\right), \]

\[ F(p_2,N)\sim Q(p_2,N). \tag{2} \]

Let us note here that the sequence \(p_1+p_2^{p_3}\) differs essentially from the sequence \(p_1+p_2^m\), \(m=1,2,\ldots\), studied in papers \((^1,^3,^4)\), and for this latter sequence, generally speaking, theorems analogous to Theorems 1 and 2 do not hold.

§ 2. We outline the proof of Theorem 2. Denoting by \(R(n,p_2)\) the number of solutions of equation (1), we estimate the quantities \(Q(p_2,N)\) and \(F(p_2,N)\) from below. We have

\[ Q(p_2,N)\geqslant F(p_2,N) =\sum_{n\leqslant 2N} R(n,p_2) -\sum_{\substack{n\leqslant 2N\\ R(n,p_2)\geqslant 2}} R(n,p_2) \geqslant \]

\[ \geqslant \sum_{n\leqslant 2N} R(n,p_2) -\sum_{n\leqslant 2N} R(n,p_2)\{R(n,p_2)-1\}. \tag{3} \]

Let \(S(n)\) be equal to the number of solutions of the equation \(n=p_i-p_j\) in prime numbers \(p_i,p_j\), under the condition \(p_i\leqslant N\), \(p_j\leqslant N\), and let us use identity (2) of the paper of N. P. Romanov \((^1)\). This identity in the present case can be

can be written in the form

\[ \sum_{n\leqslant 2N} R^2(n,p_2) = \sum_{n\leqslant 2n} R(n,p_2) +2\sum_{1<n=p_2^{q_1}-p_2^{q_2}\leqslant N} S(n), \tag{4} \]

where \(q_1,q_2\) run through the prime numbers.

From (3) and (4) it now follows that

\[ Q(p_2,N)\geqslant F(p_2,N)\geqslant \sum_{n\leqslant 2N} R(n,p_2) - 2\sum_{1<n=p_2^{q_1}-p_2^{q_2}\leqslant N} S(n). \tag{5} \]

In what follows let \(c,c_1,c_2,\ldots\) denote certain absolute constants.

We turn to estimates of the sums entering into inequality (5). According to the well-known theorem of Viggo Brun—Schnirelmann \((^2)\),

\[ S(n)<c\,\frac{N}{\ln^2 N}\,f(n), \qquad f(n)=\prod_{p\mid n}\left(1+\frac1p\right), \tag{6} \]

where \(p\) runs through the prime divisors of \(n\).

On the basis of estimate (6) we obtain

\[ \sum_{1<n=p_2^{q_1}-p_2^{q_2}\leqslant N} S(n) < c\,\frac{N}{\ln^2 N} \sum_{q_2<q_1\leqslant \ln N/\ln p_2} f\!\left(p_2^{q_1}-p_2^{q_2}\right) = \tag{7} \]

\[ = c\,\frac{N}{\ln^2 N}\,f(p_2) \sum_{q_2<q_1\leqslant \ln N/\ln p_2} f\!\left(p_2^{q_1-q_2}-1\right) \leqslant \frac32\,c\,\frac{N}{\ln^2 N} \sum_{q_2<q_1\leqslant \ln N/\ln p_2} f\!\left(p_2^{q_1-q_2}-1\right). \]

Next put \(h=q_1-q_2\), where \(q_2<q_1\) are prime numbers and

\[ q_2,q_1\leqslant \ln N/\ln p_2. \tag{8} \]

Then for every \(h\) satisfying \(1\leqslant h\leqslant \ln N/\ln p_2\), again applying the result of Viggo Brun’s “sieve”—estimate (6), we find that the number of solutions of equation (8) does not exceed

\[ c\,\frac{\ln N}{\ln p_2\cdot \ln^2 \frac{\ln N}{\ln p_2}} \left\{ \max_{h\leqslant \ln N/\ln p_2} \prod_{p\mid h}\left(1+\frac1p\right) \right\} < c_2\, \frac{\ln N\cdot \ln\ln\ln N}{\ln p_2\cdot \ln^2 \frac{\ln N}{\ln p_2}}. \]

Therefore from (7) we obtain

\[ \sum_{1<n=p_2^{q_1}-p_2^{q_2}\leqslant N} S(n) < c_0\, \frac{N\ln\ln\ln N}{\ln N\cdot \ln p_2\cdot \ln^2 \frac{\ln N}{\ln p_2}} \sum_{1\leqslant h\leqslant \ln N/\ln p_2} f(p_2^h-1). \tag{9} \]

To estimate the last sum we introduce the notation: \(\delta(k,p_2)\) is the exponent to which the number \(p_2\) belongs modulo \(k\), \(\mu(k)\) is the Möbius function, \(\varphi(k)\) is Euler’s function, \(l=\delta(d,p_2)\).

We now get

\[ \sum_{1\leqslant h\leqslant \ln N/\ln p_2} f(p_2^h-1) < c_4\sum_{1\leqslant k\leqslant N}\frac{\mu^2(k)}{k} \sum_{\substack{1\leqslant h\leqslant \ln N/\ln p_2\\ p_2^h\equiv 1\pmod{k}}}1 < c_5\,\frac{\ln N}{\ln p_2} \sum_{1\leqslant k\leqslant N} \frac{\mu^2(k)}{k\,\delta(k,p_2)}. \]

The series on the right, as N. P. Romanov \((^1)\) proved, converges. Moreover, if one puts

\[ \sigma(l,p_2)= \sum_{\substack{p_2^l\equiv 1\pmod d,\; l\mid \varphi(d)}} \frac{\mu^2(d)}{d}, \]

then the following inequalities hold:

\[ \sum_{1\le k\le N}\frac{\mu^2(k)}{k\delta(k,p_2)} < c_6\sum_{l=1}^{\infty}\frac{\sigma(l,p_2)}{l} < c_7\ln^2\ln 2p_2. \]

The proof of the last of these inequalities is rather complicated and can be obtained by the method of paper \((^1)\).

Thus we find:

\[ \sum_{1\le h\le \ln N/\ln p_2} f\left(p_2^h-1\right) < c_8\frac{\ln N\cdot \ln^2\ln 2p_2}{\ln p_2}. \]

Combining this estimate with inequality (9), we obtain

\[ \sum_{1<n-p_2^{q_1}-p_2^{q_2}\le N} S(n) < c_9 \frac{N\ln\ln\ln N\cdot \ln^2\ln 2p_2} {\ln^2\frac{\ln N}{\ln p_2}\cdot \ln^2 p_2}. \tag{10} \]

To estimate the first sum in (5), introduce the notation: \(\pi(x)\) is the number of primes \(\le x\), and \(P(x)\) is the number of numbers \(p_2^{p_3}\le x\).

Applying the prime number theorem, we shall have

\[ \sum_{n\le 2N} R(n,p_2) = \pi(N)P(N) = \frac{N}{\ln p_2\cdot \ln\frac{\ln N}{\ln p_2}} + O\left(\frac{N}{\ln p_2\cdot \ln N}\right). \tag{11} \]

Collecting the estimates (5), (10), and (11), we obtain

\[ F(p_2,N) \ge \frac{N}{\ln p_2\cdot \ln\frac{\ln N}{\ln p_2}} + O\left( \frac{N\ln\ln\ln N} {\ln p_2\cdot \ln^2\frac{\ln N}{\ln p_2}} \right). \tag{12} \]

On the other hand, obviously we have

\[ F(p_2,N)\le Q(p_2,N)\le \sum_{n\le 2N} R(n,p_2), \tag{13} \]

so that from (10)—(13) we finally obtain formula (2). Theorem 2 is thereby proved.

§ 3. Similar considerations also make it possible to obtain the following nontrivial generalization of the results of paper \((^4)\).

Theorem 3. Let \(J_k(a,N)\) be the number of numbers \(n\le 2N\) representable in the form

\[ n=p+a^m, \tag{14} \]

where \(p\) is prime; \(a\ge 2\) is a given integer; \(m=1^k,2^k,\ldots;\ k\ge 2\) is an arbitrary fixed integer; \(p\le N;\ a^m\le N\). Let \(G_k(a,N)\) be the number of those numbers which are representable in the form (14) in only one way. Then we have:

\[ G_k(a,N)\sim J_k(a,N), \]

\[ J_k(a,N)= \frac{N}{(\ln N)^{1-1/k}\ln^{1/k} a} + O\left( \frac{N}{\ln^{1-\varepsilon}N\cdot \ln^{1/k}a} \right), \]

where \(\varepsilon>0\) is an arbitrarily small constant.

Tashkent State Pedagogical Institute
named after Nizami

Received
16 XII 1957

CITED LITERATURE

\(^1\) N. P. Romanov, Uspekhi Mat. Nauk, vol. 7, 47 (1940).
\(^2\) L. G. Shnirelman, Uspekhi Mat. Nauk, vol. 7, 7 (1940).
\(^3\) E. Landau, Acta Arithmet., 1, 43 (1935).
\(^4\) A. F. Lavrik, Dokl. Akad. Nauk SSSR, 115, No. 3, 445 (1957).