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To the memory of S. Sidon. Let 0 < a, < a,. .. be an infinite sequence of positive integers. Denote by f(n) the number of solutions of n=a i +a;. About twenty years ago, SIDON 1) raised the question wether there exists a sequence a; satisfying f(n) > 0 for all n > 1 and lim f(n)'nE = 0 for all t > 0. In the present note, I will construct such a sequence. In(More)
Recently Littlewood and Offord1 proved the following lemma Let x1, x2, .-. , x, be complex numbers with I x1I ? 1. Consider the sums Fr_, erxr , where the ek are ± 1. Then the number of the sums ~r_,etx, which fall into a circle of radius r is not greater than cr2n(log n)n-1-1. In the present paper we are going to improve this to cr2 7 zii 1 / 2. The case(More)
A phylogenetic tree, also called an ''evolutionary tree,'' is a leaf-labeled tree which represents the evolutionary history for a set of species, and the construction of such trees is a fundamental problem in biology. Here we address the issue of how many sequence sites are required in order to recover the tree with high probability when the sites evolve(More)
THEOREM. Each set of 2n-1 integers contains some subset of n elements the sum of which is a multiple of n. PROOF. Assume first n = p (p prime). Our theorem is trivial for p = 2, thus henceforth p > 2. We need the following LEMMA. Let p > 2 be a prime and A = {a,, a 2 ,. . ., a,} 2 5 s < p a s tegers each prime to p satisfying ca, a 2 (mod p). Then the set a(More)
Define f (k, l) as the least integer so that every graph having f (k, 1) vertices contains either a complete graph of order k or a set of l independent vertices (a complete graph of order k is a graph of k vertices every two of which are connected by an edge, a set of I vertices is called independent if no two are connected by an edge). Throughout this(More)
We describe how deletion-correcting codes may be enhanced to yield codes with double-strand DNA-sequence codewords. This enhancement involves abstractions of the pertinent aspects of DNA; it nevertheless ensures specificity of binding for all pairs of single strands derived from its codewords—the key desideratum of DNA codes– i.e. with binding feasible only(More)
In the present paper I discuss some problems in number theory which I have thought about in the last few years ; computational techniques can be applied to some of them. 1. On Prime Factors of Consecutive Integers Let f (k) be the smallest integer with the property that the product of f (k) consecutive integers all greater than k is always divisible by a(More)
The construction of evolutionary trees is a fundamental problem in biology, and yet methods for reconstructing evolutionary trees are not reliable when it comes to inferring accurate topologies of large divergent evolutionary trees from realistic length sequences. We address this problem and present a new polynomial time algorithm for reconstructing(More)