Maximal sets of integers not containing k + 1 pairwise coprimes and having divisors from a specified set of primes

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1 EuroComb 2005 DMTCS proc. AE, 2005, Maximal sets of itegers ot cotaiig k + 1 pairwise coprimes ad havig divisors from a specified set of primes Vladimir Bliovsky 1 Bielefeld Uiversity, Math. Dept., P.O , D-33501, Bielefeld, Germay, vbliovs@math.ui-bielefeld.de We fid the formula for the cardiality of maximal set of itegers from [1,..., ] which does ot cotai k + 1 pairwise coprimes ad has divisors from a specified set of primes. This formula is defied by the set of multiples of the geeratig set, which does ot deped o. Keywords: greatest commo divisor, coprimes, squarefree umbers 1 Formulatio of the result Let P = {p 1 < p 2,...} be the set of primes ad N be the set of atural umbers. Write N) = {1,..., }, P) = P N). For a, b N deote the greatest commo divisor of a ad b by a, b). Let S, k) be the family of sets A N) of positive itegers which does ot cotai k + 1 coprimes. Defie f, k) = max A. A S,k) I the paper [1] the followig was proved. Theorem 1 For all sufficietly large where f, k) = E, k), E, k) = {a N) : a = up i, for some i = 1,..., k}. 1) Let ow Q = {q 1 < q 2 <... < q r } P be fiite set of primes ad R, Q) S, 1) is such family of sets of positive itegers that for the arbitrary a A R, Q), a, ) r q j > 1. I [2] was proved the followig Theorem 2 Let r q j, the f, Q) = max A = max M2q 1,..., 2q t, q 1... q t ) N), 2) A R,Q) 1 t r where MB) is the set of multiples of the set of itegers B c 2005 Discrete Mathematics ad Theoretical Computer Sciece DMTCS), Nacy, Frace

2 336 Vladimir Bliovsky I [2] the problem was stated of fidig the maximal set of positive itegers from N) which satisfies the coditios of Theorems 1 ad 2 simultaeously i.e. which is a set A without k + 1 coprimes ad such that each elemet of this set has a divisor from Q. This paper is devoted to the solutio of this problem. I our work we use the methods from the paper [1]. Deote R, k, Q) S, k) the family of sets of positive itegers with the property that a arbitrary a A R, k, Q) has divisor from Q. For give s ad T = {r 1 < r 2 <...} = P Q let F, k, s, Q) R, k, Q) is the family of sets of squarefree positive umbers such that for the arbitrary a A F, k, s, Q) we have r i, a) = 1, i > s. For give s, r cardiality of the family F, k, s, Q) ad cardialities of each A F, k, s, Q) are bouded from above as. Next we formulate our mai result which extet the result of the Theorems 1, 2 ad i some sese iclude both of them. Theorem 3 If Q, the for sufficietly large the followig relatio is valid ϕ, k, Q) = max A = max MF ) N), 3) A R,k,Q) F F,k,s 1,Q) where s is the miimal iteger which satisfies the iequality r s > r. 2 Proof of the Theorem 3 Let s remid the defiitio of the left pushig which the reader ca fid i [2]. For the arbitrary a = up α j, p i < p j, p i p j, u) = 1, α > 0 ad p j Q or p i, p j Q 4) defie L i,j a, Q) = p α i u. For a ot of the form 4) we set L i,j a, Q) = a. For A N deote { Li,j a, Q), L L i,j a, A, Q) = i,j a, Q) A, a, L i,j a, Q) A. At last set L i,j A, Q) = {L i,j a, A, Q); a A}. We say that A is left compressed if for the arbitrary i < j L i,j A, Q) = A. It ca be easily see that every fiite A N after fiite umber of left pushig operatios ca be made left compressed, L i,j A, Q) > A ad if A R, k, Q), the L i,j A, Q) R, k, Q). If we deote O, k, Q) R, k, Q) the families of sets o which achieved max i 3) ad C, k, Q) R, k, Q) is the family of left compressed sets from R, k, Q), the it follows that O, k, Q) C, k, Q). Next we assume that A C, k, Q) O, k, Q).

3 Sets of itegers without k + 1 coprimes ad with specified divisors 337 For the arbitrary a A we have the decompositio a = a 1, where a 1 = r α1, r i < r j, i < j, = q β1 ; q jm < q js, m < s; α j, β j > 0. If a = r α1 q β1 A, α j, β j > 0,the ā = r i1... r if q j1... q jl A as well ad also â = ua A for all u N : ua. Cosider all squarefree umbers A A ad for give the set of all a 1 such that a 1 A. This set is the ideal geerated by the divisio. The set of miimal elemets from this ideal deote by P, A ). It follows that A O, k, N)), A = M{a 1 ; a 1 P, A )}) N), For each we order {a 1 1 < a 1 2 <...} = P, A ) lexicographically accordig to their decompositio a 1 i = r... r if. Let ρ is the maximal over the choice of positive iteger such that r ρ divide some a 1 or which a 1 i a2 A. From the left compressedess of the set A it follows that a = a 1 j a2, j < i also belogs to A. The the set B of elemets b = b 1 b 2, ad a 1 i b1, a 1 j b1, j < i is exactly the set Deote ad Ba) = u : u = rα1 b 1, r q j ) rρ αρ q β1 F ; α i, β i > 0, = 1 such that b 2 = q β1, β j > 0 F, P ρ, A ) = { a P, A } ) : a, r ρ ) = r ρ, s Ps ρ A ) = a P ρ, A ) for some, such that, L ρ ) = a P ρ,a ) Ba). r j q j q j = q s = 1. The the set r s=1 P s ρ A ) is exactly the set a P ρ, A ) of umbers which are divisible by r 2 ρ. Because each a P, A ) for all has divisor from Q it follows that for some 1 s r Ba) 1 r L ρ ). 5) a P ρ s A ) Next for this s we defie the trasformatio where P, A ) = P, A ) P ρ, A ) ) R ρ s, A ), R ρ s, A ) = { v N; vr ρ P ρ s, A) }, P ρ s, A ) = { a = a 1 P ρ s A ) }. It is easy to see that P, A ) S, k, Q).

4 338 Vladimir Bliovsky Next we prove that if r ρ > r, the ) M P, A ) N) > A 6) which gives the cotradictio to the maximality of A. For a Rsa ρ 2, A ), a = r i1... r if q j1... q jl, r i1 <... < r if < r ρ, q j1... q jl = deote Da) = v N) : v = rα1 i 1 q β1 T, α j, β j 1, T, r j q j = 1. It ca be easily see that ad Da) Da ) =, a a M P, A ) P ρ, A ) )) Da) =. Thus to prove 6) it is sufficiet to show, that for large > 0 α i,α,β i 1 Da) > r Bar ρ ). 7) To prove 7) we cosider three cases. First case whe /ar ρ ) 2 ad ρ > ρ 0. It follows that Bar ρ ) c 2 r α1 rρ α q β1... q β 1 1 ) l r j = c 2 r i1 1)... r if 1)r ρ 1)q j1 1)... q jl 1) At the same time Da) = v N); v = r... r if q j1... q jl F 1, F 1, 1 1qj ) r j q j 8) 1 1 ) ) 1 1qj. r j = 1 Da) ad we obtai the iequalities Da) Da) c ) ) 1 1qj. 9) r i1... r if q j1... q jl r j Thus from 8), 9) it follows that Da) Bar ρ ) c 1 c 2 r ρ r i1 1)... r if 1) r i1... r if c 1 c 2 f ) 1 1rj r ρ j [r] {,..., } ) 1 1qj > r. 1 1qj )

5 Sets of itegers without k + 1 coprimes ad with specified divisors 339 Now let s /ar ρ ) 2, ρ < ρ 0. The we obtai the iequalities Bar ρ ) < 1 + ɛ) r i1 1)... r if 1)r ρ 1)q j1 1)... q jl 1) Da) > 1 ɛ) r i1 1)... r if 1)q j1 1)... q jl 1) From these iequalities it follows that Da) Bar ρ ) > 1 ɛ 1 + ɛ r ρ > r. 1 1 r j ) 1 1 ) ) 1 1qj, r j 1 1qj ). Here the last iequality is valid for sufficietly small ɛ because r ρ > r. The last case is whe 1 /ar ρ ) < 2. I this case Bar ρ ) = 1. Let s r i1... r if r ρ q j1... q jl = Bar ρ ). The we choose r g > q j1 ) r ad > q r j r q j. We have r ρ > r g. Ideed, otherwise > which is the cotradictio to our case. Hece g r j q j > 2 q j > 2ar ρ {r i1... r if q j1... q jl, r i1... r if q 2... q jl,..., r i1... r if q r... q jl, r i1... r if q j1... q jl r ρ } Da). Thus i this case also Da) > r = r Bar ρ ). From the above follows that for sufficietly large > 0 Q) for all a R ρ s, A ) iequality 7) is valid ad takig ito accout 5) we obtai 6). This gives the cotradictio to the maximality of A. Hece the maximal r ρ P Q which appear as the divisor of some a P a2, A ) such that MA ) N) O, k, Q) satisfies the coditio r ρ r. This iequality gives the statemet of Theorem. This is joit work with R.Ahlswede. Refereces [1] R.Ahlswede ad L.Khachatria, Maximal sets of umbers ot cotaiig k + 1 pairwise coprime itegers, Acta Arithm., LXXII.1, 1995, [2] R.Ahlswede ad L.Khachatria, Sets of itegers with pairwise commo divisor ad a factor from a specified set of primes, Acta Arithm., LXXV.3, 1996, [3] H.Halberstam ad K.Roth, Sequeces, Oxford Uiversity Press, 1966 [4] T.Apostol, Itroductio to Aalytic Number Theory, Spriger-Verlag, N.Y., Berli, 1976

6 340 Vladimir Bliovsky