数理解析研究所講究録
1091 巻 1999 年 233-240

233

On some additive problems with primes and almost-primes

T. P. Peneva\S 1. Introduction and statement of the result.
In 1937 I. M. Vinogradov [11] solved the ternary Goldbach problem proving that forevery sufficiently large odd integer $N$ the equation(1)$p_{1}+p_{2}+p_{3}=N$

has solutions in prime numbers.
Two years later van der Corput [3] used the method of I. M. Vinogradov and established the existence of infinitely many arithmetic progressions of three different primes.
A corresponding result for progressions of four or more primes has not been proved yet.
In 1981, however, D. R. Heath-Brown [4] proved that there exist infinitely many arithmetic progressions of four different terms, three of which are primes and the fourth is(as usualdenotes an integer with no more than prime factors, counted according tomultiplicity).
Another famous and still unsolved number theory problem is the so-called prime twins
conjecture, which asserts that there exist infinitely many primes , such that $p+2$ is alsoa prime. The most important achievement in studying this problem is due to Chen [2].
In 1973 he proved that there exist infinitely many primes , such that $p+2$ is .
In 1997 D. I. Tolev and the author [8] applied the Hardy-Littlewood circle method andthe Bombieri-Vinogradov theorem as well as some arguments belonging to H. Mikawa,and proved that there exist infinitely many non-trivial arithmetic progressions of three
primes, such that for two of them,and , say, both the numbers $p_{1}+2,$ $p_{2}+2$ arealmost-primes.
Later D. I. Tolev [9] obtained an extension of the above result by applying the vector
sieve developed by Iwaniec [5] and used also by Br\"udern and Fouvry [1]. He establishedthat the equation$p_{1},$$p_{2},$$p_{3}$$P_{2}$$P_{f}$$r$$p$$P_{2}$$p$$p_{1}$$p_{2}$$p_{1}+p_{2}=2p_{3}$has infinitely many solutions in different primes$p_{2}+2=P_{5}’,$ $p3+2=P_{8}$.

$p_{1},$$p_{2},$$p_{3}$, such that $p_{1}+2=P_{5}$ ,Here we study the solvability of the equation (1) in primes, such that $p_{1}+2$ ,$p_{2}+2,$ $p_{3}+2$ are almost-primes. We follow the approach of [9] putting emphasis on theexamining of the main term where we apply some arguments of [1] (for the other details
the reader may refer to [9] .
Our main result is the following$p_{1},$$)$$p_{2},$$p_{3}$

234

Theorem. Suppose that $N\equiv 3$ (mod 6) is a sufficiently large integer. Then there exist
infinitely many solutions of the equation (1) in primes
such that $p_{1}+2=P_{5}$ ,$p_{1},$$p_{2}+2=P\prime 2p_{3}+=5fP_{8}$$p_{2},$$p_{3}$.

)

In fact, the proof yields that for some constant $c_{0}>0$ there are at least $c_{0}N^{2}(\log N)^{-6}$ing (1) and such that for any prime factor of $p_{1}+2$
triplets of primes$p\geqN^{0.167}$$p_{2}+2$
and for any prime factor of $p_{3}+2$ we have $p\geq N^{0.116}$ .
we haveorNotice that if $N$ is a sufficiently large odd integer, not satisfying the hypothesis of thet,hen for
. Therefore
we haveofso
by modifying slightly the given proof, we may obtain that for such $N$ the equation (1), such that $p_{1}+2=P_{6},$ $p_{2}+2=P_{6}’$ ,has infinitely many solutions in primes$p_{3}+2=P_{9}$ . Here the extra prime factor in
is 3.
$p_{1},$$p_{2},$$p_{3},$$p$$\mathrm{S}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}\mathfrak{h}^{r}$$p$$1_{11}\mathrm{t},\mathrm{i}\circ \mathrm{n}$) $p_{3}(p_{1}+2_{\text{ノ}})(n_{)}‘+2)\backslash A\sim’(A\gamma)3^{+2_{\text{ノ}}})$$.3|p_{\mathrm{A}}\rceil p_{\sim}’$$(])\backslash \text{ノ}$$\mathrm{T}\mathrm{h}_{\mathrm{P}0}\Gamma \mathrm{P}\mathrm{m}\wedge \text{ノ}$$\mathrm{a}\mathrm{n}_{d}\mathrm{v}$$i$$p_{1},$$p_{2},$$p_{3}$$P_{r}$

Recently H. Mikawa (unpublished result) used the theory of”well-factorable” functions(for the definition see formulas (2))
and showed that the power of $N$ in the quantity
can be taken to be equal to 4/9 instead of 1/3. This enables us to prove the Theorem
with $p_{3}+2=P_{6}$ .
We should also mention that by applying the method of [9], D. I. Tolev [10] provedthat if $N$ is a sufficiently large integer satisfying the congruent condition $N\equiv 5$ (mod 24)
then the equation$D_{3}$$p_{1^{+p_{2}+p^{2}p^{2}}}^{22}3+4^{+}p_{5}^{2}=N$

such that each of thehas infinitely many sollutions in prime numbers$p_{4}+2$$p_{3}+2$$p_{2}+2,$$p_{1}+2,$
prime
and
has at most 6
factors and $p_{5}+2$ has atnumbers
most 7 prime factors.
$p_{1},$$p_{2},$$p_{3},$$p_{4},$$p_{5}$\S 2. Notations.
Let $N$ be a sufficiently large integer, such that $N\equiv 3$ (mod 6) and$\alpha_{2}<1/4,0<\alpha_{3}<1/6$ , which we shall specify later.
ing
constants$\alpha_{1},$$\alpha_{2},$$\alpha_{3}$-$0<\alpha_{1},$$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{S}}\mathfrak{h}$We put$z_{i}=N^{\alpha_{i}}$,$\dot{i}=1,2,3$,$z_{0}=(\log N)1000$$D_{1}=D_{2}=N^{1/2}\exp(-2(\log N)^{0}\cdot 6)$$P(z_{0})= \prod_{<2p<z0}p$

,

,

,

,

$\mathrm{t}D_{0}=\exp((\log N)^{0}\cdot 6)$$D_{3}=N^{1/3}\exp(-2(\log N)^{0}\cdot 6)$,(2)$i=1,2,3$ .

$P(Z_{0}, zi)= \prod pz0\leq p<zi$ ’-prime numbers.
denote integers;
Letters $m,$$\mu(n),$M\"obius’
function,
Euler’s function and the number
denoteand
As usual
denote the greatest
andof positive divisors of , respectively;. Instead of $m\equiv n$ (mod )common divisor and the least common multiple of
. For
and
means that
we write for simplicity $m\equiv n(k)$ . The notation
instead of $A<<B\ll A$ .
positive and we write
$n,$

$d,$

$l,$

$k,$

$h,$

$\delta,$$\nu,$$t,$$p,$$\rho$$p_{1},$$p_{2},\ldots$$\tau(n)$$\varphi(n)$$[m_{1}, \ldots, m_{k}]$$(m_{1}, \ldots, m_{k})$

$n$

$k$

$m_{1},\ldots,m_{k}$$p^{\nu}||n$

$A$

$B$

$A_{\wedge}\vee B$$p^{\nu}|n$$p^{\nu+1}\parallel n$235

For squarefree odd integers$k_{1},$$k_{2},$$k_{3}$and prime$p$

we denote,(3)$I_{k_{1},k_{2},k_{3}}(N)=pi+p12 \equiv 0(ki)^{p},=1,2+p2+\sum_{i,3}\log p_{1\mathrm{g}}\mathrm{l}\mathrm{o}3=Np_{2}\log p_{3}$$|$$1/(p-1)^{3}$$-1/(p-1)^{2}$$-1/(p-1)^{2}$$1/(p-1)$$1/(p-1)$$h_{k_{1},k_{2},k_{3}}(p)=$

$-1$
$-1$
$\sim$$p-1$$\omega(k_{1}, k_{2}, k_{3})=\prod_{p|k1k2k3}\frac{1+h_{k_{1},k_{2},k}(3p)}{1+h_{1,1,1}(p)}$

if
if
if
if
if
if
if
if

,

$p\parallel k_{1}k_{2}k_{3}$$p\chi_{k_{1}k_{2}}k_{3}$$p||k_{1}k_{2}k_{3}$$p||k_{1}k_{2}k_{3}$

,
,
,
,
,

$p\parallel N$;$p|N$ ;
$p\parallel N+2$;

$p|N+2_{\}}$(4),, $p|N+4$ ;$p^{3}||k_{1}k2k3$ ,$p\parallel N+6$ ;, $p|N+6$ ;$p^{2}||k_{1}k2k_{3}$$p\parallel N+4$$p^{2}||k_{12}kk_{3}$$p^{3}||k_{1}k_{2}k_{3}$$\Omega(k_{1}, k_{2}, k_{3})=\frac{\omega(k_{1},k_{2},k_{3})}{\varphi(k_{1})\varphi(k_{2})\varphi(k_{3})}$

6 $(N)= \prod_{p\parallel N}(1+\frac{1}{(p-1)^{3}})\prod_{p|N}(1-\frac{1}{(p-1)^{2}})$ .

,

(5)

(6)

\S 3. Outline of the proof.
Consider the sum$\Gamma=(pi+2,P^{+}p12)=\sum_{N+p3=,1i=},\log p_{1}\log p_{2}\log p_{3}1,2,3^{\cdot}$

Any non-trivial estimatethat $p_{i}+2=P_{h_{i}},$$\mathrm{h}\mathrm{o}\mathrm{m}$$h_{i}=[\alpha_{i}^{-1}],$

below of$\Gamma$

implies the solvability of (1) in primes, such

$i=1,2,3$ . We see that$\Gamma=p2p\sum_{p_{1}++\mathrm{s}=N}\log p1\log p2\log p3\Lambda 1\Lambda 2\Lambda 3\Lambda 4\Lambda_{5}\Lambda_{6}$where$2, \sum_{d|(_{\mathrm{P}i+}P(z\mathrm{o},zi))}\mu(d)$for$\dot{i}=1,2,3$,$d|(p_{i-} \epsilon+2,P(\sum_{0z))}\mu(d)$for$\dot{i}=4,5,6$.

$\Lambda_{i}=\{$

Denote$d|( \mathrm{p}_{i+2},P(z\sum_{z0)i))}\lambda^{\pm}i(d)$for$d|(+2P \sum_{p_{i}-3(z\mathrm{o}))},\lambda_{0}^{\pm}(d)$for $i=4,5,6$ ;$\dot{i}=1,2,3$,,$\Lambda_{i}^{\pm}=\{$

(7)

236

are the Rosser’s weights of orderwhereparticular, we have$\lambda_{i}^{\pm}(d)$$|\lambda_{i}^{\pm}(d)|\leq 1$,

for$\lambda_{\dot{\iota}}^{\pm}(d)=0$

We find that$\Lambda_{i}^{-}\leq\Lambda_{i}\leq\Lambda_{i}^{+}$$D_{i}$,,$0\leq\dot{i}\leq 3$$d\geq D_{i}$,(see Iwaniec [6], [7]). In$0\leq\dot{i}\leq 3$.

(8)$1\leq\dot{i}\leq 6$(for the proof see [7]). Consequently we may apply Lemma 3 of [9], which is the analogue
of Lemma 13 of . and we get(9),$[1]\backslash$$\Gamma\geq\Gamma_{0}$where$\mathrm{r}_{0}=p_{1+2+}p\sum_{=p3N}\log p_{1}\log p_{2}\log p3(\Lambda_{12}-\Lambda+\Lambda^{+}\Lambda_{4}3+_{\Lambda^{+_{\Lambda^{+}}}56^{+}}$.

$+\Lambda_{1}^{+}\Lambda^{-_{\Lambda_{3}^{+}}}2\Lambda_{4}+\Lambda_{5}+\Lambda_{6}++\cdots+\Lambda_{1}^{+}\Lambda_{2}+_{\Lambda_{3}^{+_{\Lambda_{4}}}}+\Lambda_{5}+\Lambda_{6}--5\Lambda_{12}^{+}\Lambda^{+}\Lambda^{+}\Lambda^{+}\Lambda^{+_{\Lambda_{6})}}345+$

We use (3), (7) and change the order of summation to obtain(10)$\Gamma_{0}=d_{i}|P(z0\delta_{i}|P(z’ 0zi\sum_{3),i=1,2},\hslash(d1, d_{2}, d3, \delta_{1}, \delta_{2}, \delta_{3})I_{d\delta_{1}},d_{2}\delta_{2},d\mathrm{s}\delta\epsilon(1N)),i=1,2,3’$

where$\kappa(d_{1}, d_{2}, d\mathrm{s}, \delta_{1,2}\delta, \delta_{3})$

$=$

$+$

$+$

$\lambda_{1}^{-}(d_{1})\lambda+2(d_{2})\lambda 3+(d_{3})\lambda_{0}+(\delta 1)\lambda_{0}^{+}(\delta_{2})\lambda^{+}0(\delta_{3})$. . . . . ... . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .
$\lambda_{1}^{+}(d_{1})\lambda 2+(d_{2})\lambda^{+}(3d3)\lambda_{0}+(\delta_{1})\lambda^{+}(0\delta 2)\lambda_{0}-(\delta_{3})$$5\lambda_{1}^{+}(d_{1})\lambda_{2}+(d2)\lambda_{3}+(d3)\lambda^{+}0(\delta_{1})\lambda_{0}^{+}(\delta_{2})\lambda^{+}(0\delta_{3})$

.

(11)

By applying the Hardy-Littlewood circle method we find an asymptotic formula forwhich we substitute in (10). Proceeding in the same way as in Lemma
the sum
11 and Lemma 13 of [9] we derive$I_{k_{1},k_{2})k}S(N)$$\Gamma_{0}=\frac{1}{2}N26(N)W+\mathcal{O}(\frac{N^{2}}{\log^{4}N})$

where$\mathfrak{S}(N)$

,

(12)

is defined by (6) and(13)$W=d_{i}|P \delta_{i}|P(z(z0_{0},zi\sum_{3),i=1,2},\hslash(d_{1}, d_{2,3,1,2,3}d\delta\delta\delta)\Omega(d1\delta),i=1,2,31, d2\delta 2, d3\delta 3).$By (4) it is obvious that for squareffee odd integers$h_{1,1,1}(p)$$h_{p,1,1}(p)$$h_{k_{1},k_{2},k_{3}}(p)=\{$$h_{p,p,1}(p)$$h_{p,\mathrm{P}\mathrm{P}},(p)$

if
if
if
if

$k_{1},$$k_{2},$$k_{3}$

we have$p\parallel k_{1}k2k_{3;}$$p||k_{1}k_{2}k3$;$p^{2}||k_{1}k2k_{3}$;$p^{3}||k_{1}k2k3$

.

(14)

237

Define$\omega_{1}(p)=\omega(p, 1,1)$,$\omega_{2}(p)=\omega(p,p, 1)$,$\omega_{3}(p)=\omega(p,p,p)$

.

(15)

It is clear that if $p>2$ then1$\omega_{1}(p)=\{$

if$\frac{(p-1)^{2}}{p^{2}-3p+3}$$\frac{(\mathrm{p}-1)(\mathrm{P}^{-}2)}{p^{2}-3p+3}$$p|N$ ;$\mathrm{L}_{\frac{-1}{-2}}p$

if $p|N+2$ ;if$p\parallel N(N+2)$$0$$\omega_{2}(p)=\{$;$\frac{(p-1)^{2}}{p^{2}-3_{\mathrm{P}}+3}$

if$p|N$ ;

if $p|N+4$ ;if$p \int N(N+4)$ ;(16)

4$\omega_{3}(p)=\{$

if$p=3$ ;

if $p|N+6$ ,$\frac{(p-1)^{3}}{p^{2}-3p+3}$

if

$0$

$p\parallel N+6$

$p>3$ ;.

By (4), (5), (14), (15) we get$\omega(k_{1}, k2, k3)=\mathrm{P}\nu_{1\leq\leq}k_{1}\prod_{||k_{23}k,\nu 8}\omega\nu(p)$

.

(17)

The next statement is the analogue of Lemma 12 of [1]. The Lemma follows easily
from (16), (17).
Lemma 1. For squarefree odd$k_{f}$

let$\omega^{*}(k)=\prod_{kp|}\omega_{1}(p)$If$(k_{2}, k_{3})$$k_{1},$$k_{1\mathrm{z}}k_{2},$. Then.

$k_{3}$

is a triplet of(i) there exists a function
we have$k_{2},$$g$$\dot{i}ntegerS$.

, we put $k_{1,2}=(k_{1}, k_{2}),$ $k_{1,3}=(k_{1}, k_{3}),$ $k_{2,3}=$of the three variables$k_{i,j}$, such that for any squarefree odd$k_{3}$$\omega(\grave{k}_{1}, k_{2,3}k)=\omega^{*}(k_{1})\omega(*k2)\omega(*k3)g(k1,2, k_{1},3, k_{2,3})$and$g(k_{1,2}, k1,3, k_{2},3) \leq 10(\max k_{i,j})^{10}$(ii) for any squarefree odd$k_{1},$$k_{2:}k_{3}$;we have the inequality$\omega(k_{1}, k_{2,3}k)\leq 10\tilde{\omega}(k_{1})\tilde{\omega}(k_{2})\tilde{\omega}(k_{3})$

,

238

where

$\tilde{\omega}(m)$is the multiplicative function defined on squarefree odd $m$ by$\tilde{\omega}(p)=\{$

2

if $p \int N+6$ ;if $p|N+6$ .

$2p^{1/3}$

Suppose that the integers
Using (5) and (17) we easily get$d_{1},$$d_{2,3,1}d\delta,$$\delta_{2},$$\delta_{3}$satisfy the conditions imposed in (13).

$a_{2}’,$$\alpha cc_{2,33\mathrm{u}}/’\backslash \cap\prime c_{1}\mathrm{o}3\mathit{1}\iota(,$$\Omega^{\gamma}\langle d_{1}o_{1},\dot{u}_{2}Od\mathrm{o}\grave{)}=\Omega \mathrm{c}\backslash \dot{u}1,$

Note thatobtain by (11), (13)$\Omega(\delta_{1}, \delta_{2}, \delta 3)$$\delta 2,$$\circ_{3)}^{c\backslash }$is a symmetrical function with respect to$W= \sum_{i=1}^{6}L_{i}H_{i}-5L_{7}H_{7}$.
$\delta_{1},$$\delta_{2},$$\delta_{3}$. Hence, we

,

(18)

where
$L_{1}$$=$$Pi \sum_{d_{i}|(z_{0},z),i=1,2,3}\lambda_{1}-(d1)\lambda 2+(d_{2})\lambda 3+(d_{3})\Omega(d_{1}, d2, d_{3})$$L_{2}$$=d_{i}|P(z \mathrm{o},zi)i=1\sum_{2,3},,\lambda+(1d_{1})\lambda_{2}^{-}(d2)\lambda 3+(d_{3})\Omega(d1, d2, d_{3})$$L_{3}$$=$$i=1 \sum_{2,3},\lambda+(1d_{1})\lambda 2+(d_{2})\lambda^{-}(3d3)\Omega(d_{1}, d2, d_{3})$,,,$d_{i}|P(z0,zi),$$L_{4}=L_{5}=L_{6}=L_{7}$$=d_{i}|P(z0,z_{i}) \sum_{=i1,2,3},\lambda^{+}1(d_{1})\lambda 2+(d_{2})\lambda^{+}(3d3)\Omega(d_{1}, d_{2}, d3)$

$H_{1}=H_{2}=H_{3}=H_{7}$$=$$H_{4}=H_{5}=H_{6}$$=$$\delta_{i}|P(z\mathrm{o})\sum_{=i1,2,3},\lambda^{+}(0\delta_{1})\lambda_{0}+(\delta_{2})\lambda^{+}(0\delta_{3})\Omega(\delta_{1}, \delta 2, \delta_{3})$$\delta_{i}|P())i=1\sum_{z0,2,3}\lambda^{-}0(\delta_{1})\lambda_{0}+(\delta_{2})\lambda^{+}(0\delta_{3})\Omega(\delta_{1}, \delta_{2}, \delta 3)$

,

,

.

It is easy to prove the following
is a function defined on the set of integers and such, satisfying $(n_{1}n_{2}n_{3,1}ll2l_{3})=1$ , we haveand$\phi(n_{1}l_{1,2}nl2, n3l_{3})=\phi(n_{1}, n_{2}, n_{3})\phi(l_{1}, l2, l3)$ . Then thefunction

Lemma 2. Suppose thatthat for any two triplets$\phi(n_{1},n_{2}, n_{3})$$n_{1},$$n_{2},$$n_{3}$$l_{1},$$l_{2},$$l_{3}$$\Phi(n)=\sum_{nd_{1},d_{2},d\mathrm{a}|}\emptyset(d1, d2, d_{3})$is multiplicative.
Applying Lemma 1 and Lemma 2 we find asymptotic formulas for the sums
$i=1,2,3\mu(\delta 1)\mu(\delta_{2})\mu(\delta_{3})\Omega(\delta 1, \delta_{2,3}\delta)$$H^{(\mu)}= \sum_{-}\delta_{i}|P(z\mathrm{o}),$.

$H_{i}$. Define
(19)

239

Lemma 3. We have
$H_{i}=H^{(\mu})+\mathcal{O}((\log N)-10)$,$1\leq\dot{i}\leq 7$

,

and
(20)

$H^{(\mu)_{\vee}}\wedge(\log z_{0)^{-3}}$

Now we are able to estimate from below the quantity $W$ , defined by (18). We put$\mathcal{F}(z_{0}, zi)=\prod(z_{0}\leq p<zi1-\frac{\omega_{1}(p)}{p-1})$

where$\theta_{i},$$\omega_{1}(p)$,$s_{i}= \frac{\log D_{i}}{\log z_{i}}$,$i=1,2,3$ ,(21)

is defined by (16). Suppose that $c^{*}>0$ is an absolute constant and letsatisfy$s_{i},\dot{i}=1,2,3$$\theta_{1}+\theta_{2}+\theta_{3}=1$,$\theta_{i}>0$,$i=1,2,3$ ,$f(S_{i})-2\theta iF(_{S_{i})}>C^{*},$

where and are the functions of the linear sieve. Following the arguments in the proof
of Lemma 15 of [9] it is easy to establish that
$f$

$F$

$W \geq H^{(\mu)}\prod_{j=1}^{3}F(Z_{0}, z_{j})(_{i=1}\sum^{3}(f(S_{i})-2\theta iF(Si))+\mathcal{O}((\log N)^{-1/3}))$

.

(22)

Finally, we choose$\alpha_{1}=\alpha_{2}=0.167$,$\alpha_{3}=0.116$,$\theta_{1}=\theta_{2}=0.345$,$\theta_{3}=0.31$

and compute that for sufficiently large $N$ we have$f(s_{i})-2\theta iF(s_{i})>10^{-5}$,$\dot{i}=1,2,3$

Therefore, using (2), (20) $-(23)$ we get.

$W \gg\frac{1}{\log^{3}N}$

The last estimate and (9), (12) imply$\Gamma\gg\frac{N^{2}}{\log^{3}N}$,

which suffices to complete the proof of the Theorem.

.

(23)

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References[1] J. Br\"udern, E. Fouvry, Lagrange’s Four Squares Theorem with almost prime variables,
J. Reine Angew. Math. 454 (1994), 59-96.

[2] J. Chen, On the representation of a large even integer as the sum of a prime and the
product of at most two primes, Sci. Sinica 16 (1973), 157-176.
[3] J. G. van der Corput, \"Uber Summen von Primzahlen und Primzahlquadraten, Math.
.
Ann. i16$\langle 1939^{\backslash })’,$$\tau\perp-5_{\mathrm{U}}^{\cap}$[4] D. R. Heath-Brown, Three primes and an almost-prime in arithmetic progression, J.
London Math. Soc. (2), 23 (1981), 396-414.

[5] H. Iwaniec, On sums of two norms from cubic fields, Journ\’ees de th\’eorie additive desnombres, Universit\’e de Bordeaux I, (1977), 71-89.
[6] H. Iwaniec, Rosser’s sieve, Acta Arith. 36 (1980), 171-202.
[7] H. Iwaniec, A new form of the error term in the linear sieve, Acta Arith. 37 (1980),

307-320.

[8] T. P. Peneva, D. I. Tolev, An additive problem with primes and almost-primes, Acta
Arith. 83 (1998), 155-169.
[9] D. I. Tolev, Arithmetic progressions of $pr\dot{i}me-almoSt-pr\dot{i}me$ twins, to appear in Acta

Arithmetica.

[10] D. I. Tolev, On the representation of an integer as a sum of five squares of prime
numbers of a special type, preprint of Plovdiv University ”Paisii Hilendarski”, 1998.
[11] I. M. Vinogradov, Representation of an odd number as a sum of three primes, Dokl.
Akad. Nauk SSSR 15 (1937), 169-172 (in Russian).

Institute of Mathematics
University of Tsukuba
Tsukuba-shi
Ibaraki 305

JAPAN
E–mail: tpeneva@hotmail.com