Note on Normal Numbers Arthur H. Copeland and Paul Erdös

A Note on Normal Numbers in Matrix Number Systems

We consider a generalization of normal numbers to matrix number systems. In particular we show that the analogue of the Theorem of Copeland and Erdős also holds in this setting. As a consequence this generalization holds true also for canonical number systems.

Note on Normal Decimals

It was proved by Champernowne [2] that the decimal . 1234567891011 . . . is normal, and by Besicovitch [11 that the same holds for the decimal . 1491625 . . . . Copeland and Erdös [3] have proved that if p i, P2 . . . . is any sequence of positive integers such that, for every 0 < 1, the number of p's up to n exceeds nB if n is sufficiently large, then the infinite decimal •p ip 2p 3 . . . is n...

The Extension of an Arbitrary Boolean Algebra to an Implicative Boolean Algebra

An Extension of Dhh-erdös Conjecture on Cycle-plus-triangle Graphs

Consider n disjoint triangles and a cycle on the 3n vertices of the n triangles. In 1986, Du, Hsu, and Hwang conjectured that the union of the n triangles and the cycle has independent number n. Soon later, Paul Erdös improved it to a stronger version that every cycle-plus-triangle graph is 3-colorable. This conjecture was proved by H. Fleischner and M. Stiebitz. In this note, we want to give a...

On the Least Primitive Root of a Prime Paul Erdös and Harold N. Shapiro

Lastly, Erdös [2] proved that for p sufficiently large (1 .5) g(p) p' 2(log p) . This last result, of course, is not directly comparable with the others, giving better results for some primes and worse results for others . In any event, all of the results are very weak (as is evidenced by a glance at tables of primitive roots [1]) in relationship to the conjecture that the true order of g(p) is...