In [2] it was shown that A(&, 4) = co for fe^ 1048909 and it was conjectured that A(&, 4) = =° for all k. In this paper we establish this conjecture with the following Theorem. A(&, 4) = ». Proof. It suffices to prove the theorem for values of k which are prime. The proof makes use of the following proposition which is a special case of a result of Kummer [l] (see also [3]). Proposition. Let k ...

The Greek mathematician Diophantus of Alexandria studied the following problem: Find four (positive rational) numbers such that the product of any two of them increased by 1 is a perfect square. He obtained the following solution: 1 16 , 33 16 , 17 4 , 105 16 (see [4]). Fermat obtained four positive integers satisfying the condition of the problem above: 1, 3, 8, 120. For example, 3 · 120+1 = 1...

A collectionC of distinct 3-element subsets of the set S = {1, 2, . . . , n} is called an identifying system of the quadruples of S, if every 4-element subset of S contains at least one of the triples in C, and moreover, no two 4-element subsets of S contain exactly the same triples of C. We consider the minimal possible cardinality of such a system, and show, in particular, that for n = 6, 7 a...

We show that reconstructing a tree from order information on triples is NPhard. This is in contrast to the case for ultra-metrics and for subtree information on quadruples which are both known to allow polynomial time reconstruction.

We show that there are only finitely many Diophantine quadruples, that is, sets of four positive integers {a1, a2, a3, a4} such that aiaj +1 is a square for all 1 ≤ i < j ≤ 4, consisting of Fibonacci numbers.

We extend our assessment of the potential of perturbative coupled cluster (CC) expansions for a test set of open-shell atoms and organic radicals to the description of quadruple excitations. Namely, the second- through sixth-order models of the recently proposed CCSDT(Q-n) quadruples series [J. J. Eriksen et al., J. Chem. Phys. 140, 064108 (2014)] are compared to the prominent CCSDT(Q) and ΛCCS...