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1 Math-Net.Ru All Russian mathematical portal U. V. Linnik, On the representation of large numbers as sums of seven cubes, Rec. Math. [Mat. Sbornik] N.S., 1943, Volume 12(54), Number 2, Use of the all-russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use Download details: IP: December 26, 2018, 09:35:22

2 1943 МАТЕМАТИЧЕСКИЙ СБОРНИК Т. 12 (54), N. 2 RECUE1L MATHEMATIQUE On the representation of large numbers as sums of seven cubes U. V. Linnik (Leningrad) 1 In the present paper we prove that G(3)<7, i.e. every sufficiently large number is the sum of 7 non-negative cubes. This is an improvement of a result of E. Landau [1]: G(3)<8. Our proof is based on some elementary identities as well as on our previous results on the theory of positive ternary quadratic forms. In particular, we use the following Theorem [2]. Every positive ternary quadratic form F(x,y,z) with odd invariants [2, 1] such that there exists a prime a>/2 with f ) = (-^-) represents all sufficiently large numbers that are prime to 22 and satisfy the genus conditions of F*. In particular, such forms that f j = f^ J for any o/2 are called «convenient» [2]; they represent also every even number ms/=0 (mod 4) satisfying the conditions: (m, 2) = 1, f ^\ = C^^), m > c 0 (2). As one can prove by the methods of [2], the reciprocal to convenient forms f(x,y,z) have genus conditions modulus 8 only and represent, in particular, every number m=/e0(mod 4), for which the congruence m = /($, *r), C) (mod 8) is solvable and m = l (mod 5), when m>c Q (Q) and 2=1 (mod 5). Such forms will be used here**. Such are, in particular, the forms / (x, y, z) = A?x 2 + A\*f + A?z 2, where Д 15 Л 2, A z are primes satisfying the conditions: and т 19 т 2, T 3 are odd; A t^a 2^^A z^\ (mod 5). * As for G. Pall remarks (Math. Rev., November 1941), see [4]. ** To prove it by the methods of [2] the following quaternion equations should be considered: b + L = Q- P - X where L 2 = Qm; Norm P = Q; Norm Q = 5^ * b s ± 1 (mod 5); b = 0 (mod 2).

3 Representation of numbers as sums of cubes We shall use the following identity Щ + Щ + Щ xl + yl+xl + yl+xl + y\ [Н г Q^-Щу + Нъ (x 2 - ЩУ + Н г (Х 3-3 ) 2 }, where H i = x i + y i (/=1, 2, 3). Thus, if N x satisfies the equality N = H + Щ-РН1 + 3 [н г (x, - Щy + H 2 (x 2 - Щy + H z (x 3 - Н^У), (2) where х г are integers and Н г are even, then N x is equal to the sum of 6 cubes xl + yl + xl + yl + xl + yl (3) where Уг = Я* x* "(i=l, 2, 3). 3 The set of numbers that are sums of 6 (7) positive cubes will be denoted by a 6 (respectively o 7 ). Lemma 1. If N satisfies (2) with additional conditions H^ К О ~ш)~ 3 ' N " 0 +ieo) i ] ( r ' = 1 > 2 > 3 )' W Proof. We obviously may assume that in (2) x t ~ > 0 and, since Я; > 0, also Xi>0. To obtain y i = H i x t^0 it suffices to show that Xi^Hi or х*-^*<^\ We have 2 ^ 2 Я 4 >0, 3tf,(x,-^), <W-"» +? + H '<N- 3W^' 4 1 ' 1 ') О*" 2)* = N(i + ^. юоу' «"О- )* I * 2 I ^ 2 " Lemma 2. From any three progressions contained in the progression 4H + 1 it is possible to choose three numbers p 1? p 2, p 3, and from any 3*

4 220 U. V. Linnik three progressions contained in a progression of the form 4n \ the primes Я if 12, Q 3 c an be chosen, satisfying the conditions: ( )- (Й)- 1 - (8)-: ( 0 =1; '> /=l ' 2 ' ъ > (5) Proof. As it is known, the existence of primes q x, q 2, q 3 satisfying the last two conditions follows immediately from Dirichlet's theorem on progressions. The numbers p t satisfying the conditions of the second group form a system of progressions, from which they can be chosen in such a way that the first conditions will be satisfied. Lemma 3. // A 1 = p??i 1 ', A 2 = p;*q, A z = pl*ql*', A and q> being defined above, then every form A lx *+A 2 y 2 + Aa\ (6) where % x, i[, т 2, т 2, Ч 3, т 3, are odd, is reciprocal to a convenient form (according to the terminology used in [2]), /. e. for any prime (o/a h ; k=l, 2, 3. Proof. fmilmi\ J_ i _ (~ x \ The other conditions can be verified analogously by means of the reciprocity law. In particular, (Т=)= =- - =Ш=-'=("1!)- We remark here that if p 4, p b, p 6 is another system of numbers р г then the same conditions hold for the form pppl'x 2 + Pl 2 Pl 2 'y 2 + Pl*pl*'z with V <, *., <, * 8 > < odd, if (g) = + l, a = 4, 5, 6; *=1, 2, 3. Lemma 4. Let r\ be arbitrarily small. For sufficiently large B>0 iften? <?x/s/ /шля f {x, y, z)=d 1 x 2 4-D a y 2 + ^3z 2 satisfying the conditions: 1 - / (*> У? 2) is reciprocal to a convenient form, thus it represents every m> c 0 (DfiJ)^; m =/= 0 (mod 4); (m, D X D 2 D Z ) == 1; m == 1 (mod 5); m = /(x, у, z) (mod 8). 2. Z) 1? D 2, D 3 //e ш /fte segment [(1 ч2в, В]. 3. D 1 = D 2 =D 3 = 1 (mod 15). 4. D 15 D 2, D 3 are //ft r 0/ /fte form 8n + 3 or 8n + 7 simultaneously, or of the form 8n + \ or 8n + 5 also simultaneously- Proof. Our proof will be based on a theorem of B. Segal [3] on the distribution of numbers of the form a x b y. From this theorem follows,

5 Representation of numbers as sums of cubes 221 in particular, that if a and Ь are distinct primes then for the neighbouring numbers M t und M i+i of the form a 2 " x ' +1 the following estimation is valid: Mi+i-M^oiMt+i). (7) Let three progressions contained either in 4n + l or in 4n 1 simultaneously be given and YJ>0 be fixed. We take the forms considered in Lemma 3 fl?ft? x 8 +fl?ft?'y a + ai*bi*'z*, (*) where z b i\ are odd, such that a t and b L satisfy the conditions: 1) а^еее^еее 1 (mod 15); 2) ajbi (i = l, 2, 3) are either of the form 8n + 3 or 8n + 7, or of the iorm 8n + l or 8n + 5 (every а { Ь ь can be of each form of these two groups separately). 3) a 1 b 1 x 2 + aj) 2 y 2 + ^z z2 are forms considered in Lemma 3. Then, since д^==й?'==1 (mod 15) and a?= a t (mod 8), b} if ~ Ь г (mod 8), every (*) is of the required form. Consider the aggregates of numbers Ajiftji' == M i? e?ft?' = N t, aj«6?' = P*. (8) Choose К > 0 such that for the given r\ > 0 If В is sufficiently large then there are numbers of each kind (8) bet- Ft ween В and B' = n 2b2 ^. By B. Segal's estimation (7) we ob~ tain: Let M, N, P be of the form (8) and lie in the segment [В', B]. Then the corresponding right neighbours of them belong to the same segment and are at the distance less than -тт- from M 9 N and P. If they still lie in the segment [В', В] then the distances between them and their right neighbours are also less than -^-. Consequently, В and -к + В are numbers, the existence of which is stated in Lemma 4. 4 We put 1 f}=(\ ~r 0 Y and take В > sufficiently!large in order that there exist between (1 TJ) В and В the representatives of all possible forms modulo 8 considered in Lemma 4. These representatives are EEEX 2 + y 2 + z 2 (mod 15) and are respectively of the form ±(x 2 + y 2 + z 2 ), ±3(x 2 + y 2 '+z 2 ), _-(x 2 + ); 2 ) + 3z 2, x* + y* + bz\ 5(-x 2 -y 2 + 3z 2 ) modulo 8. Denote by A the product of all their coefficients.

6 222 U. V. Linnik If N sufficiently large is given, we choose a prime p satisfying the conditions 1) P 2B 3 2) p = a I (mod 5Л); 3) p = a2 (mod 3); 4) 3p^ = 1 (mod 8). "К' + ту 1 гв ъ If D lf D 2, D 3 are any numbers between fl\ # f = A 2/7 (f = l, 2, 3) then we shall have i i i -утшу б 3 and В 3 (9) and For any ЛГ, such that we shall have by (9) Я,б [W r (l 1000J ' ^ C J J* (10) (11) 5 Let p be fixed corresponding to the given N. Since p^=2 (mod 3), there exists, as it is known, С such that ЛГ-С 3 =0 (mod 3p), 0< <3p. Every number of the form С = С + Л 3p satisfies this congruence, as soon as С does. Since p ^N s /10 20, we have for 0</c<10 8, for instance: i V ^ ^ 4 ^ 1Л80 ^ 1 П and thus, when iv^n-c' 3, N t will satisfy (11). Therefore if (2) holds for a certain N ± =N^^B then N o 7. Let the representatives of forms considered in 4 be /i(x,y,z),..., M*^ *); s<64 (12) (their number does not exceed 4 3 = 64). It follows from (2) and (11) that if an equality of the form N-V*=(Dl t + Dl t + Dh)2p* + 6pf t (p, ij, C) (13) where / f (p, 7], g = D lt p 2 + D 2t Y] 2 + D 3t C 2, l</<3, is valid, then N a 7. to virtue of the choice of the forms (13) and of their properties point-

7 Representation of numbers as sums of cubes 223 cd out in 1, it is sufficient to prove that the numbers /cg[0,10 8 ] and t < s can be chosen such that 1. JV-C' 3 is even. 2. ^ D b + Dl t+ D hp^n,. s pr. me tq A 3. iv' = l (mod 5). (14) 4. N either is not divisible by 4 and satisfies the congruence N = f t (x, y, z) (mod 8) or is of the form AN" with ЛГЕЕЕО (mod 4) and N" = f{?y T), C) (mod8) [we notice that N» = U (p, ъ С) implies 4N» = f t (2 P, 2 4f 2Q]. In virtue of the condition (9), 2) imposed on p we obtain: If M is arbitrary then at least П f 1- J Ю 8 > 67 6 of numbers of the form tlzjp-.m; C' = C + fc-3p; 0<k<10 8 ; are prime to A. )3. _1_ )3. -l-/)3. ll Furthermore, all ^ 3 2 * ^ 3' p 2 are integers [since D it EEEl (mod 15)] and =EE1 (mod 5), because p=l (mod 5). 6 Consider first the case N = 2 (mod 4). We take m 1^-x- (mod 8). Such one exists, since N is even and 3p is odd. m x is odd and is of one of the forms 8/z+l, 8Л + 5, 8Л + 3, 8Л + 7. Consider the case ш 1 = 8Л+Ь We put C' = 4C" and take к such that (14), 3 is satisfied. Then^ = 0 (mod 8). We choose, further, the form / t (x, y,z) (x 2 + y 2 + z 2 ) (mod 8) and, having we choose к varying it modulo 20 in order to get: {N', A) = l. We shall have ^ ^ 3 ~ M = N' = 1-(-1)=EE2 (mod 8). The congruence N'EEE2=E: (x 2 -by 2 -fz 2 ) (mod 8) is solvable, consequently, as we^know, if N is sufficiently large, then Ngo 7. The other cases considered analogously can be given by the following table 6p (mod 8) f t(x, y, z) (mod 8) N' (mod 8) 1. = 1 2. =5 3. =3 4. ~7 s (x 2 + y 2 + z 2 ) s (x a + У 2 + z 2 ) ss (x 2 + y 2 -f z 2 ) ~ (x a + y 2 + z 2 ) H-Ds2 5Ч-1)з6 3 1 ==2 7-1 =6 The congruence N' ss и (mod 8) is satisfied

8 224 Ю. В. Линник 7 If N is odd then C is odd too. We have, further, N Z' Z ~N~ N ' (mod 8) and it is possible to take С so that '-^-g-^ is odd. We come back to our table, where should be replaced by N ~ g ". Let, finally, N be even and N = 2*N 0, (N 09 2)=1. Without loss of generality we may suppose that a = 2, since N 0 G 7 implies 2 3 W 0 ga 7 and the case a=l has been studied above. If N = 2 2 (2h+\) = 8h + 4 then, since 3p=l N (mod 8), бр- :2 (mod 4) and is of the form 8Л + 2 or 8Л + 6. Putting C'=4C", we obtain ~ p = ~ (mod 8). The cases, when -^- is of the form 8h + 2 or 8Л + 6, will be given by the table: /Qfr ~ (mod 8) ft (x, y, z) (mod 8) AT (mod 8) The congruence N' EZ f f (mod 8) 1. ~=2 2. =6 s (x 2 + y 2 -f z 2 ) s (x 2 + y 2 + z 2 ) =5 is satisfied»» Thus, all possible cases are investigated and we obtain that every N' sufficiently large belongs to o 7. References [1] E. Landau, Vorlesungen uber Zahlentheorie, 1927, Bd. II. [2] U. Linnik, Bulletin Acad. Sci. URSS, 4, 4 5, [3] B. Sega 1, С R. Acad. Sci. URSS, ] U. Linnik, C. R. Acad. Sci. URSS, 1942, in print. (Поступило в редакцию 25/IV г.) О разложении больших чисел на семь кубов Ю. В. Линник (Ленинград) (Резюме) В настоящей работе доказывается, что G (3) < 7. Для доказательства используются результаты автора по теории тернарных квадратичных форм и работа Б. И. Сегала о распределении чисел вида а х Ь у ; (а, Ь) = 1. Развитая автором теория представления больших чисел отдельной тернарной формой «удобного» типа [2] опирается на неэффективную теорему С. L. Siegel^ о числе классов в k(\f m). В данной работе ее можно было бы заменить более слабой теоремой Е. Landau о количестве чисел т с h( m) = o(^ ^ доказательство. ^ и получить эффективное