170 P. ERDŐS, r- FREUD ad N. HEGYVÁRI THEOREM 3. We ca costruct a ifiite permutatio satisfyig g ilog log ( 4) [ai, ai+i] < ie c yio i for all i. I the

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1 Acta Math. Hug. 41(1-2), (1983), ARITHMETICAL PROPERTIES OF PERMUTATIONS OF INTEGERS P. ERDŐS, member of the Academy, R. FREUD ad N. HEGYVARI (Budapest) For the fiite case let a1, a 2,..., a be a permutatio of the itegers 1, 2,..., ad for the ifiite case let a 1, a2,..., ai,... be a permutatio of all positive itegers. Some problems ad results cocerig such permutatios ad related questios ca be foud i [2] (see i particular p. 94). I [3] the desity of the sums ai+ai +1 is estimated from several poits of view. I the preset paper we shall ivestigate the least commo multiple ad the greatest commo divisor of two subsequet elemets. First we deal with the least commo multiple. For the idetical permutatio we have [ai, ai+1]=i(i+l). We show that for suitable other permutatios this value becomes cosiderably smaller. First we cosider the fiite case (1) THEOREM 1. We have mi max [ai, ai+1] = (1+o(1)) í1 2 1sis_1 4log where the miimum is to be take for all permutatios a 1, a2,..., a. Oe might thik that the mai reaso for ot beig able to get a smaller value lies i the presece of the large primes (see also the proof). Theorem 2 shows that this is oly partly true. THEOREM 2. Omit arbitrarily g()=o() umbers from 1, 2,..., ad form a permutatio of the remaiig oes. The. for ay fix E>0, ad large eough we have (2) mi max [aj, ai+1] >- 2-E Iti---g()-1 O the other had,.for ay e()-->o we have with a suitable g()=o() (3) mi max [a.. ai+1] < 2-8() IsiS - g() - 1 log {mi I-i max) 1 [ai, ai+1 A equivalet form of Theorem 2 is : ] must log ted to 2 for ay g()=o(), but it ca do this from below arbitrarily slowly for suitable g()=o(). I the ifiite case we obtai a much smaller upper boud : Acta Mathematica Hugarica 41, 1983

2 170 P. ERDŐS, r- FREUD ad N. HEGYVÁRI THEOREM 3. We ca costruct a ifiite permutatio satisfyig g ilog log ( 4) [ai, ai+i] < ie c yio i for all i. I the opposite directio we ca prove oly a very poor result : THEOREM 4. For ay permutatio (5) lim`sup [a i, ai+i],_ 1-l0g2-3,26. Very probably this lim sup must be ifiite, ad oe ca expect a eve sharper rate of growth. Cocerig the greatest commo divisor oly the ifiite case is iterestig. THEOREM 5. We ca costruct a ifiite permutatio satisfyig 1 (6) (ai, ai+i) 2 i for all i. (7) O the other had, for ay permutatio The right value is probably 2, lim if (ai, ai+i), i i but we could ot yet prove this. Proofs ai PROOF OF THEOREM 1. for which First we show that ay permutatio must cotai a 2 [ai, ai+i] - (1 +0(1)) 4log Cosider the primes betwee 2 ad, the umber of these is about 21 Hece ' at least oe of them has a left eighbour '(1 I o(1 )) ' ad thus the least 21 commo multiple here is -- (1 +0(1)) 'Z 2 log 2 Now we costruct a permutatio satisfyig 2 (8) [ai, ai+i] {1+0(1)} 41og for all i-1. The idea is to take the multiples of a prime p as a block, ad to separate the blocks by "small" umbers. The the l.c.m. will ot be too large at the border of the Acta Mathemattca Hugarica 41, 1983

3 PERMUTATIONS OF INTEGERS 171 blocks. Ad iside a block (9) [ai, a,+,] -- P which is good if p is ot too small. Fially we have to arrage the umbers havig oly small prime factors. Let us see the details. For the primes p up to let kp be the miimal expoet for which qp = p k p 4log (i.e. q p =p if p-4log). We ow defie the set S of the "small" separator umbers : take Ij () - ffi} ) umbers from 1 to some L just leavig out the values qp ad 2q p. Obviously L= 1z -(1+ (1)) log We start the permutatio by writig dow alterately the primes betwee ad 2 i decreasig order ad the first elemets of S i icreasig order. (Here a block cosists of p aloe.) To show (8) we observe that whe we arrive to p- e, the we have used up H()-11(c)-(1-c) log small umbers, i.e. the l.c.m. of p ad its eighbour is (10) c(1-c) to For the primes betwee 2 ad ji we slightly improve the costructio. We take the largest prime, isert all its multiples (up to ) after it, leavig its double to the ed. Now we choose the ext eve umber of S as separator, start the ext block with the double of the ext prime, put i all the multiples ad termiate it by the prime itself. The we isert the ext odd umber of S as separator ad repeat the alogirthm. (9) ad (10) show that (8) is satisfied. We ote that for p _ - we do ot have to be so careful about the parity of the separator umber, ad for p- `~ we do ot 4 vlog really eed separators at all. Next we proceed similarly with the qp values betwee }fi ad 4 log, but here of course we take oly those multiples of qp which have ot yet bee used up (either i the blocks, or as separators). qp ad 2qp lie at the two eds of a block (they were excluded from S to be ow at disposal), hece we ca either omit the separators, or put i arbitrarily large umbers as separators. We shall isert as separators the umbers still left, i.e. which have all their prime power factors less tha 4 log (ad which were ot i S). There are at most - 4log4 g 2*(41og) Such umbers, where ]j*(x) deotes the umber of prime-powers up to x sice there are 11(V)>1 /' - E blocks for 41og-gp -{fi, we ca cosume as Acta Mathematica Hugarica 11, 1983

4 172 P. ERDŐS, R. FREUD ad N. HEGYVÁRI separators all the umbers left. We have obviously 2gp -_ [ai, ai+1] og Fqp at the border of the blocks products cosistig of v such (distict) primes where v=log +4 log log. By the lemma we ca arrage these products so that ay two subsequet terms should difiside. PROOF of THEOREM 2. To prove (2) we observe the well-kow fact that there are e umbers up to which have a prime factor greater tha III I c=(1 +0 (1)) log - 1 ), hece we must keep early all of them. Whe we jump from 1 - e/2 a multiple of a large prime to a multiple of aother large prime, the we either jump directly, but the the l.c.m. is at least (l-`i2)2, / or we isert a small umber as separator, but the we eed at least (1 +0(1)~ separators, ad so we obtai a l.c.m. log greater tha 1- /2 log To prove (3) we keep oly those umbers whose largest prime factor lies tetwee 1( ) ad `( ") It is well kow that we omitted just o() umbers (see e.g. [1]). We start the permutatio by the largest prime left ad its multiples, the we put the ext prime followed by its multiples, etc. Here -~ ~ E(") for two multiples of the same p [ai, ai+i] = p I- `( " ), whe jumpig to a ext prime. PROOF OF THEOREM 3. First we ote that it is eough to costruct a permutatio a I, a2,..., of a subsequece of the atural umbers which satisfies (4), sice we ca isert the remaiig elemets afterwards arbitrarily rarely ito this permutatio. We shall use the (probably well-kow ad early trivial) statemet of the followig lemma : LEMMA. Let H be a fiite set, IH I =h ad t -h. The we ca order the subsets havig exact,, t elemets so that A H, +, Í = t--1 holds for all i. PROOF OF THE LEMMA. We prove by iductio o h. The iitial step is obvious. Now assume that the assertio is true for h-1 ad for all t. Cosider ow h ad ay t. We fix a elemet x o, take first all subsets cotaiig xo ad the take the other oes. Both parts ca be ordered suitably by the iductio hypothesis for h -1, t --1, ad for h-1, t, resp. We have o difficulty either at joiig the two parts, sice if a "good" order exists, the a simple bijectio of H ca trasform it ito aother "good" order with a prescribed first (or last) subset. The costructio of the permutatio rus by a iterative process. Assume that for some ad k=" 9 " we have a,, a2,..., a k ready ad o oe of them has a prime factor greater tha 2. We take ow all primes betwee 2 ad, ad form all the y1 Acta iimathematica Hugarica 41, 1983

5 PERMUTATIONS OF INTEGERS 173 fer oly i oe prime factor. This arragemet will be the ext segmet of the permutatio from a c}2. For a trasitio elemet ak+i we ca take e.g. ay prime betwee 2 ad. For i--k+l clearly We have formed about [ai, i+l] - v+i - ke2ylogkloglogk = i e 2ylogiloglogi (11) r = 2log log +4 log log ew terms of the permutatio thus we arrived at least to ar. The algorithm will work if (12) r > (2) log2 holds. Estimatig the biomial coefficiet i (11) as a power of the smallest factor the umerator ad the greatest factor i the deomiator we obtai log +4 log log to -41o to _ 21og g g g log+4loglog r- log + 4 log log j (lo g3 ll ) ad (12) follows by a easy calculatio. PROOF OF THEOREM 4. First we give a very simple proof of weaker form (5) with Z istead of 1-l0g 2 ' i.e. that o permutatio ca satisfy i of (13) [a i, ai+i] < l2 -s) i with a fix e for i -- io. (14) We use the iequality which is equivalet to (15) [ai, ai+i] 3 í ai a,+i _ ai a, +, 3 (ai, ai+i) + (a,, ai+i) ad hece it is obvious, sice the miimal value of the two terms o the right-had side of (15) is 1 ad 2. Assumig (13) we obtai 1 1 (2 ) 1 - (2 +E' -' +E' to -K. ) g it, [ai, ai+i] =io [ai, ai+i] 3 =,,i 3 O the other had, usig (14) we have ) i=, [ai, ai-*i] 3 i=, ai + ai+i 3 i 3 log+k which is a cotradictio if is large eough. Acta Mathematics Hugarica 41, 1933

6 174 P. ERDőS, R. FREUD ad N. HEGYVARI Now we tur to the proof of (5). Assume idirectly that for some permutatio, s>0 ad io we have 1 (16) [ai, a i+i ] < i if i -- io. 1 -lo s This clearly implies also ai i l-log2+s for i io hece a l, a2i..., a are all smaller tha (17) i N = I -log2-e if is large eough. From ow o we shall cosider oly the a,-s with i-. Let us call the primes greater tha VN ad smaller the N "large primes". If a ; ad a;+, have differet large prime factors, the [ai, ai+1 ]--N i cotradictio to (16). Hece we must isert "separators" betwee ai -s cotaiig differet large prime factors (the separators caot have large prime factors, of course). If a i is the greatest separator elemet ad ai+, has a large prime factor the [ai, a i+1]-a,vn. Hece we agai arrive at a cotradictio by showig that there are at least fn separators, or equivaletly, there are at least VN large primes which occur as factors of ai -s. We kow that there are (1 +o(1))n log 2 umbers up to N havig a large prime factor ad we have (I -log 2+e)N ai - s [see (17)], hece at least EN ai - s have a large prime factor. All of these ai - s caot be multiples of less tha V large primes : ideed, the umber of multiples up to N of VN large primes is f N N l+0 0(1))Nlo,,these P" l P g VW, p - N E/2 P - ( ( )) 1 s < rn. 1/2 PROOF OF THEOREM 5. The permutatio 1, 2, 6, 3, 12, 4, 20, 5, 35, 7,... clearly satisfies (6), i.e., if we have already costructed a, ad k is the smallest umber which was ot yet used, the a,+, should be a commo multiple of a 2 ad k (e.g. the smallest oe still available) af put a2, +2=k. To prove (7) we observe first the followig facts : Let b l, b 2,... be arbitrary differet atural umbers ot greater tha. The : (b, b2) i, (19) (b,, b2) or (b 2, 3 bs) = 3, (20) Imi (b i, bi+,) = 4. (18) is obvious. To show (19) assume idirectly that e.g. 3 <d=(b,, b 2)-(b2, b 3). The either b,=d ad b 2 =2d or b,=2d ad b 2 =d, but i both cases b 3 must be Acta Mathematics Hugarica 41, 1983

7 PERMUTATIONS OF INTEGERS 175 at least 3d which is a cotradictio. We ca show (20) by similar methods. Put d = mi (bi, bi+i) _ (bk, bk+1) 1si~4 Arguig agai idirectly we have {bk, bk+i} {d, 2d, 3d}, ad takig step by step the eighbourig umbers, ad havig i mid that the greatest commo divisor must be at least d, we obtai that r {bl,..., bj s {d, 2d, 3d, 3 2 d which is a cotradictio. Now we are ready to prove (7). Assume that (ai, ai+l ) > 1 i if i is large eoc ugh. The {a,, a 2i..., acj {1, 2,..., } if is large eough. O the other had, takig ac,,,,,..., a, _ i, a,,,, at mos tevery secod umber ca be less tha or equal to, sice 1 i 1 e c c 2 2 which is impossible by (18) if both ai ad a i+l are less tha or equal to. Similarly, usig (19) we obtai that at most the two third part of ac13,..., ac 12 is ot greater tha, ad fially usig (20) we coclude that at most the 4/5 part of ac14,..., ac 13 is smaller tha. Hece 1 61 c c 90' -- - c). (e - 2 as asserted. 1 (c _ c. 2 + (c - c), 4 c -'- -~ ) Z - REMARKS. L We ca improve (7) somewhat, if we use further iequalities of the type (18), (19) ad (20). But this does ot seem to give a serious reductio, ad also the discovery of the proper iequalities is ot too easy. E.g. mi b b ad here 12 caot be replaced by 11, as show by the umbers 81j, 162j, 108j, 54j, 216j, 72j, 144j, 48j, 96j, 192j, 128j, 64j ( = 216j, d = 48j = 9 ). 15ís12 ( ~,,+1) ` 5, 2. We metio the followig related problem, where we ca determie the extremum exactly THEOREM 6. mi {a i, j a i +, -a i j} 3 (21) lim if is true for ay permutatio, ad we ca costruct a permutatio where equality holds. Acts Mathematics Hugarica 41, 1983

8 1 7 6 P. ERDŐS, R. FREUD ad N. HEGYVÁRI : PERMUTATIONS OF INTEGERS PROOF OF THEOREM 6. The followig permutatio shows the possibility of equality : 1, 2, 3, 6, 4, 8, 5, 10,..., i.e. we always take the smallest umber still available followed by its double. To prove (21) we assume idirectly that there is a permutatio satisfyig (22) 3 ai > ( 4 +r i ad Jai},-ail > +E/ i. for i _- i o with a fix E>0. (3T The all the umbers up to (4 + E / N must occur amog a,,., an, if Nis large eough. This also meas that at least ( + e / N umbers smaller tha 4 (4 + e N must appear 111 N amog an12+1,..., a N. Thus we obtai a i~ 2, for which both a i ad a i+, are smaller tha +E)N. Say a i},>-ai, the (4 (4+EJN ai+1= ai+(ai+,-ai)>2(4+eli>2(4 +E_2 J which is a cotradictio. We ote that the proof gives slightly more, sice we did ot make really use of the e i (22). Ackowledgemet. The authors express their gratitude to Prof. M. Simoovits for this remarks cocerig Theorem 1. Refereces [1] N. G. De Bruij, O the umber of positive itegers =x ad free of prime factors zy, Idag, Matt., 13 (1951), [2] P. Erdős, R. L. Graham, Old ad Nex Problems ad Results I Combiatorial Number Theory, Moographic N 28 de L'Eseigemet Mathématique (Geéve, 1980). [3] R. Freud, O some of subsequet terms of permutatios, to appear i Acta Math. Acad. Sci. Hugar. (Received September 22, 1981) MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES BUDAPEST, REÁLTANODA U , H-1053 EŐTVÖS LORÁND UNIVERSITY DEPARTMENT OF ALGEBRA AND NUMBER THEORY BUDAPEST, MÚZEUM KRT. 6-8, H-1088 BUDAPEST, BUDAI NAGY ANTAL U. 4, H-1137 A^_1a Matiae7txatica Hugarica 41, Szegedi Nyomda - F. v. : Dobó József igazgató