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SOME PROBLEMS AND RESULTS ON CONSECUTIVE PRIMES by P. ErdL)s (Syracuse) and A. Rknyi (Rudspest) 5 1. Let $1 = 2, +, = 3, $s e 5, +,, . . . ##, . . . denote the sequence of primes. In what follows we shall be concerned with some problems regarding difference and quotient of consecutive primes. We put 1c= 2,3, . * . . There are many unsolved problems regarding the sequences d, and Q,,, and the subject is full of peculiar difficulties. For instance it has been conjectured by many authors that d+, = 2 for infinitiveIy many values of H, i.e. that the sequence of “twin primes”: 3,5 5,7 Il.13 17,19 29,31 41,43 ‘.. is infinite. Neither this, nor the more feeble conjecture that dn does not tend to infinity, has been proved up to now. Moreover at present we are unable to prove even that In this direction Z&t - an = 0 . W+COh.C~ it has been proved by P. ERD& r) that Recently R. A. RANKIN 4) proved, that for C we can choose #+, and according to an oral communication of A. SELBERG he can improve this result by choosing for C the value #. This is all we know about the small values of &. In the opposite direction it can be proved by the method of V~GGO BRUN that twin primes and more generally pairs of consecutive primes the difference of which is equal to a fixed (even) integer 2R, are rather “few”. More precisely, we have the following result *): The number of solutions of 48 = k, $,, 5 x does not exceed ** 1 (3) *) From the prime number theorem it follows only d r; 1. MI A n--too 1% n **) We denote by c,, c,, . . . positive constants, and by c a positive constant which is not always the same. 116 It follows from (3) that the sequence d,, is not bounded. As a matter of fact, this follows also from TCHEBYSHEFF’S estimation ‘) 6 (x) = dz log sb, < c,x, h.6% and also from the trivial remark that no one of the consecutive numbersti!+2,7a!+3,n!+4,...n!+-+zisprime,andthus there are arbitrarily large “gaps” in the sequence of primes. From TCHEBYSHEFF‘S estimation, mentioned above, it follows also that F dn 1 According to the primes number theorem we can sly”, log 15, z X’ choose c, = I + E for any E > 0, if x is sufficiently large and thus d, we obtain lim 2 1. In his paper &) where previous results N-MO 1% h are quoted, P. ERD& proved, that dn > 0. log &J . log log $H (1% 1% 1% %4* This has been still improved by RANK~N 8, by adding in the numerator of the denominator in (4) the factor loglogloglog fi= (4) nFa Q 2, If the conjecture regarding twin primes would be proved, it would follow that the sequence G?,,oscillates between 2 and arbitrary large values, thus it would be neither monotonically increasing nor decreasing, from some point onwards. This result has been proved without any hypothesis, by P. ERD~S and P. TURAN ‘). They proved also that the sequence q,, is also neither monotonically increasing nor decreasing. These results can be expressed also by saying that the sequences p,, and log p, are neither convex nor concave from some point onwards. A generalization of this result, regarding the sequence log fin, has brcn given recently by A. RENYI *). His result formulated in a geometrical terminology, runs as follows: Let us consider a finite polygonal line situated in the complex z-plane and having the consecutive vertices z,, z,, . . . zn. The total curvature of this polygonal line is defined by (5) where arg 2 denotes the argument of the complex number 2, i.e. if Z = r . &, r > 0; - 7~< v I; 7t, we have arg Z = v. NOW let TZNdenote the polygonal line with the vertices 2%= n + i. log &I, &+l SNandlet GN d enote its total curvature, It is evident, that 117 if the sequence log #+,would be convex or concave from some point onwards, Gn7would be bounded. Rut RBNYI proved that GN tends to infinity, as a matter of fact he proved GN > c. logloglog N, thus giving a new proof of the theorem of ERI~S and TURAN mentioned above. In waht follows we shall prove a refinement of this result, giving the exact order of magnitude of GN: TIteorem 1. If GIV denotes the total curvature, defined by (5), of the polygonal line WV having the vertices zr, = it + i. log &, p,+l 5 N, we have cg . log N < GN < c, , log N. (6) lim - GN exists, but we are at prcscnt N i-CDlogN not able to prove this, The pro01 of Theorem 1 will be completely elementary. Besides the estimations of TCHEBYSHEFF It seems probable (‘1 that *) c -5 < 72(x) < cg -q log f 410gx and #,&+I < 2p,,, WC shall use only the method of BRUN. 5 3. Proof of Theomn 1. Clearly we have (8) Using GN =+ +4; N 1arc tg log qH+l -----arc tg log qn I. n this gives arc tg a -- arc tg b = arc tg G log %!k! q* X arc tg qn+1 . log pn ’ 1 + log &+I 6 N As according to (7) the sequence qn is bounded, we obtain from (9) (f-9 GN= (10) where C,LN<GN<LN As regards the upper estimation, it can be finished at once: we have Now, we obtain *) n (x) denotes the number of primes which are $ x. 118 Thus the upper estimation of (6) is proved. As regards the lower estimation it is somewhat more intricate. We proceed as follows: us neglect the terms log y of LN for those indices it for I I which either 4, or c&,+1exceeds I Iog A,*) and denote the remaining sum by LN’; evidently GN > c LN > c LN’. As according to (7) Let p$further 1 ) <I, andwehavefor~xJ<l)log(l+x))>c,Ixl, using the identity 9n-b1 -= (14 * +a+14 9% we obtain, k4lrtn --p- ha by putting (where 2:’ denotes that the summation is extended only over such indices n for which both dn and &+I are < 1 log fi) we obtain LN > c7 (DN - (16) RN) Now we have Thus if we prove DN 2 c8 log N, GN > E%log N follows. To estimate DN we need the following lemma. Lemwuz I. The number of solutions of C = a, dn+r = b, P, < N, does not exceed In other words, the number of such primes p I; N for which both p + a and 9 + a + b are also primes, does not exceed (18). This lemma can be proved easily by the method of VIGGO BRUN (l.c.) by applying a ,,triple sieve”. We need further Lemma 2. Let us denote w (4 =b,( 1+ f). (19 ----. *) The value of the constant .A shall be determined later on. 119 We have c (20) W(m) $;B+log(A ASmsA+B Proof of Lmma + B) + I. 2. Clearly 2 ASm6A-bB ASdr5; A+B and thus ASmSA+B #?Y w (m)5 Now we prove, using Lemmas I and 2, the following Lemmu 3. Let E, I and M denote positive constants, and let. k denote a positive integer, k 1 kO(M, c, A), The number of indices S, for which cl, < IM . (k -- I), k+i < XM (k -- - I) further eMk are valid, does not exceed cm. E . eMk Mk ’ Proof of Lemm 3. The number of primes P, for which CM(A--1) < p,, < eMk, further G!,,= a and &+I = b can be estimated by Lemma I. If this number is denoted by &,b, we have C@Mk W Zasb-= @jp (a) W (a + b). If Zb denotes the number of primes satisfying Lemma 3, we obtain all conditions of for R > k, (M, E, il). Let us proceed now to the proof of our theorem. According (7) if we choose to (22) Zk < 2 a<AM(R-l), &,b. b<hM(k-1) lb-al < 7 Using Lemma 2 we obtain 120 C&Mk we have at least -2Mk primes in the interval(eM(k-11, cMR),As we have dN < eMk the number of primes P, in (@(k-1), e”) ,Wk-1) :b<&fk ,$A does not exceed AM (k--l)’ and we have the same estimate for the number of primes P, in the same interval for which dn+l 2 ;uM (R - 1). Thus for at least for which d, 2 AM (k--l), &k), WC have both dn 5 UM (k--l) and primes in (&P-i), dpl+l < AM (k-l). According to (23) the number of primes in (#(k-l), eMk) for which di < AM (k--l), &+I < AM (k-l) and &kM does not exceed ‘g, Let us choose now R= p*+l--&tI < --j=- 32 c and E = &. c eMk I . We obtain, that for at least kk primes m (eMF:-11, eMk) we zve EkM jdn+wZ+T. d, < ctM (k-i), &+r < AM (k-l), and Thus from (15) we obtain 1da-t1 --dn I> 5 2s’ (24) DN > 2C 4ilM P* kD<k < W N ,Wk-1 1<p,,<$fk -F As it has been remarked L: I >c,, 1ogN. k,<k<‘%$! above, this proves our theorem. 5 3. In connection with the above estimations there arises the question, how many times it occurs for P, 5 N that d~+i = dn, or more generally that &+l = 4, + h, where h is a fixed integer (positive, negative or zero}. In this direction we prove Theorem 2. The number of solutions of $+, I; N, u&+-r = & -t- h does not exceed (25) Proof of theown 2. We have according to Lemma 1 121 Applying the inequality of we obtain CAUCHY-SCHWARZ, HN<- ca w (m). logs%s, (logs)%+* (27) Row we need the following simple Lemma 4. ,#;I wa W < crs-4. (28) Proo/ 01 Lenama 4. We have I e 1 n” ==z If p=a ( P=S 1 + p ( P) ( I+$ 1 pr+ 2$ 1 =c11 and thus where V {d) is the number of different prime factors of d. Denoting by G (d) the number of divisors of d we have and thus Thus we obtain $Ws(m) m-l <$A which proves Lemma 4, As & (log IV)% can evidently occur 4: not more than N (log iV)-‘h times, as it is seen from 2 dn 5 N, ASN Theorem 2 is proved. $ 4. We prove still some results concerning the sequence dn. Recently W. SIERPINSM proved the following theorem: *) For every positive integer K there exist an infinity of primes + with the property that all the numbers $ -f R, k = 1,2, . . . K are composite i.e. that there are an infinity of primes which are “isolated” from both sides from the next prime by an interval of arbitrary large length. Evidently the theorem of SIERPINSKI is = 0. The proof given by SIERPINSKI (‘7 122 is based on the application of the theorem of DIRICNLET, that in every arithmetic progression Dx + Y, x = 1, 2, . . . D,Y = I there are an infinity of primes. In what follows we shall show that a theorem which is stronger than that of SIERPRINSKI can be proved in an elementary way by using only BRUN’S method. We shall prove Theorem 3. For any integer N and any Y < &log prime &, I; N for which N there is a 1% N j==O,1,2,,..(r-I). dH-i 2 Cl6 _____ra ' Proof of theorem 3. We need the following Lemma 5. 2 W (m) < 3A1-A for O<A< (I -a) 21 m=l d (291 1. We have namely which proves Lemma 5. Now WC Provo Lemma 6. (30) We apply the theorem of Vrcco We obtain using also (7) I (31) p$ a 5&N hSN -’ (,0&A RERUN mentioned in $ 1 (see (3)). i+ anA n (N) ’ ,,& ,:g’N W (4 ’7 + (lo;N: +A’ Thus by Lemma 5 WC obtain from (31) (32) 1 r; 3c N (log N)r-A -A) pl:N ad - logsNil{ = qi-- 3c N --a) (log N)‘+A’ which proves Lemma 6. Now Theorem 3 can be deduced easily. Let us put 123 T-l 1 c j=l dG’ We have by Lemma 6 with ,I= 4. 8(‘) n = (33) c hsN-r-i-1 (34) Thus if we put (7) that Atin 8:) = ,fi’,) we obtain hS N-r+1 (35) A:r’ < ;gN* Thus for an appropriate it we have (36) dw-j > Cl6 1% N Y3 from (34) using for i=O,l,...(r-.-1), which proves Theorem 3. For Y = 2 this thcorrm states, that there exists a constant C such that for every N there is a prime fi 5 N for which all the numbers 9 f k, k = 1, 2, , . [Clog N] are composite. This can be formulated also by saying, that the first prime having the propcrty required by the theorem of SIERPINSKI, i.e. for which all numbers + f k, k = 1, 2, . . I K are composite, does not exceed eCK. We still add some remarks on Lemma 6. As it has been used CN before, for more than primes & 5 N we have d, < K log N, log N and thus we obtain (37) This means that the order of magnitude of the sum on the left of (30) given by Lemma 6 is exact. Now we can make the parameter A in this sum depend on N, for example we may choose A=l.-- 1 . loglog N’ We obtain from Lemma 6 that (38) cl,Nlogrog N c ‘< p*SN 46 logsN--’ For the sum on the left of (38) we can give only the lower estimate (39) ArN’ 1 z fi ’ %l!DN (log N)” 124 As amatter of fact, if we could prove a% that dim = 0, but at n-cc0 logfi present we ate unable to prove anything of this type. To give some idea for the random distribution of the sequence A, we add a tabel giving the values of ~6, for P, S, 4397, (n = 2,3, . . . 599). with y(N) Budapest, -+ 00, it wou1.d follow January 30, 1949. References: 1. P. EIZDOS, The (1940) 448-441. 2. R. A. RANKIN, 4. 5. 6. 7. 8. 0. of consecutive primes, Dnke Math. Journal, The difference between consecutive prime numbera, III, Sot. Journal, 22 (1947) 228230. VIGGO BRUN, Le crible d’Eratosthene et le th&oreme de Goldbach, Videnskapselskapets Skrifter, I, No. 3, Kristiania, 1920. P. L. TCHEBYSHEFF, Memoire sur les nombres premiers, Journal de math. pures et appl. I. ser. 17, (1862) SW-3!l0 pp. P. ERD&, On the difference of consecutive primes, Quarterly Journal of Mathematics, Oxford ser. 0, No. 22, (1936) 124-128 pp. R. A. RANKIN, The difference between consecutive prime numbers, London Math. !%oc. Journal, 13 (1938) 242-247. P. ERD~S and P. TuRhN, On some new question on the distribution of prime numbers, Bulletin Amer. Math. Sot. 54, No. 4, 1948.371-378 pp. A. R~NYI, On a theorem of Erdiis and Turan, Bulletin Amer. Math. Sot. in print. W. SIERPINSKI, Remarque sur la repartition dea nombres premiers, Colloquium Mathematicum 1 (1948) 193-194. London 3. difference Math. 125 n 0 t 10 238 40 50 00 70 iFi 100 IO 28 30 40 50 60 70 80 on 300 10 20 30 40 60 i;a ;;i 6a 300 IO 20 3a 40 60 (iC TC 80 4z 16 2c 3c 41 6C Ii0 70 80 Table of the sequence d, for IZ = 2,3, , I . 599. 0, 1, 2, 3, 4, 6, 6, 7, I 0 4 19 Q 2 4 2 8 12s 2 0 : 2 lb 4 6 i u 1: 10s t 4 4 t 4 4 4 2; u 2 0 1; 240 10 ::: 1: 10 : u 1: 8 1: 0 1: 2 6 6 8 1; 14 1: 10 2 a 2” 10 12 12 i : 12 12 1: 2 8 II 14 I3 4 2 11 ; 12 t 12 4 6 (i 14 1: 1: 11 12 12 $ 20 30 40 60 60 1: 18 22 2 Rcl ii 12 8 ii t - 4 4 8 10 2 a E 4 ; 12 20 1:: 10 2e 1: 4 t 2 8 i 28 4 0 2 6 : 4 t 1: 12 4 : : 4 2: 2: :28 12 8 1: 2: 14 2 1: 11 I8 4 2 2 4 2 1; ii 8 1:: la8 6 8 1: i 1: 8 12 1; lb 6 2 2 4 1: 4 : 4 6 1: 1: ;: 1: 2 1: ; : 1: 1: 2 1: 6 6 : 1; 1: 2 8 4 14 1: 14 2 8 11 2 0 0 4 6 2 : 10 14 t 1: 4 4 2 6 tl 1: t 10 6 : 2 8 1: 2 2 4 0 6 2 6 4 I(i 6 2 8 I6 6 4 t 6 4 2 2; 4 2 2 4 4 1: 1: 10 18 2 2 G 1: 8 0 1: 4 t *t ii 4 1: ; 6 6 1: IO 2 2 2: 2 8 :f 10 D 3:: 8 2 6 4 :: : 1: 10 4 2 1: 6 8 “6 8 1; :t 8, -2 1: II: 12 4 :t 0 t 4 4 6 ::: 4” 2 20 6 6 0 4 f 122 10s 1: lb 10 2 1: II 1: 12 6 fi 2 2 4 4 i 6 4 II 2 4 22 1: 24 4 0 8; 2 1: 4 8 1; 2 6 2: E i 22 10 8 11 8 2 1: 11 : 4 4 22 2 i 10 1: 18 :t 12 4 8 f 20 2 I:, 6 0