ON SOME APPLICATIONS OF GRAPH THEORY TO NUMBER THEORETIC PROBLEMS DEDICATED TO THE MEMORY OF

Let a1 <. .. < ak < n be a sequence of integers no one of which divides any other ; then it is easy to see that [1] max k = [n 2 11. On the other hand, I proved [2] by a combination of number theoretic) and graph theoretic methods that, if we assume a1 I ajak i 0 k), then ((T(n) denotes the number of primes < n.) r(n)+c l n2/3 /(log n) 2 < max k <<T(n)+r2n2 / 3/(log n) 2. .. (1) Further, if we only assume that the products aiaj are all distinct, then [2] r(n)+C3n3/4/(log n) 3/2 < max k < T(n) +C4n 314 (2) In the present paper we prove that the lower estimation in (2) is sharp (apart from the value of the absolute constant c 3). In fact we prove the following THEOREM. Assume that a 1 <. .. < ak < n is a sequence of integers for which the products ajal are all distinct. Then 1r(n)+C3n 3 /4 /(log n) 3/ 2 < max k < w(n)+c5n 3 /4 /(log n)3/2. The proof of (3) will be similar to that of (2) and will use elementary results from number theory and graph theory. Before we prove our Theorem we would like to discuss a few related results. An old conjecture of Turin and myself states as follows : Let b1 <. .. be an infinite sequence of integers. Denote by f (n) the number of solutions of n = bi+b5. Then, if f (n) > 0 for all n > no , lim sup f (n) = oo. Probably the above conclusion n-oo also follows if we only assume that bk < c e k 2 for all k. These conjectures are not yet settled and are probably quite deep. It is perhaps surprising that the multiplicative analogies of these conjectures have been settled. In fact I proved the following results [3] : Let a 1 <. .. be an infinite sequence. Denote by g(n) the number of solutions of n = aia j. Then, if g(n) > 0 for all n > n o , we have lim sup g(n) = oo. In fact the following stronger result holds : n-> co Assume that a 3 <. .. < ak < n, n sufficiently large and k > (1+e)n(log log n)1-1/(l-1) ! log n. Then for some …