The number of factorizations of numbers less than $x$ into divisors greater than $y$

ON THE NUMBER OF PRIMES LESS THAN OR EQUAL x

(3) Z I"—1 log p = x log x + 0{x). r-ix\-pJ Since [x/^>]= [[#]/£], it is clear that (3) then holds for all real x>0. We propose to show in this note that the order of w(x) =the number of primes less than or equal x (a result originally due to Tschebyschef, cf. [l]) may be derived very quickly from (3) as a consequence of a general theorem which has no particular relationship to prime numbers. T...

finite simple groups with number of zeros slightly greater than the number of nonlinear irreducible characters

the aim of this paper is to classify the finite simple groups with the number of zeros at most seven greater than the number of nonlinear irreducible characters in the character tables. we find that they are exactly a$_{5}$, l$_{2}(7)$ and a$_{6}$.

There are less transcendental numbers than rational numbers

According to a basic theorem of transfinite set theory the set of irrational numbers is uncountable, while the set of rational numbers is countable. This is contradicted by the fact that any pair of irrational numbers is separated by at least one rational number. Hence, in the interval [0,1] there exist more rational numbers than irrational numbers.

On the Number of Prime Numbers less than a Given Quantity

Prime numbers are probably one of the most beautiful objects in all of mathematics. It is remarkable, that they have such a simple definition: “p is prime iff p has no other divisors, besides 1 and p”, and at the same time their properties are so hard to explore. The importance of the primes was realized in antiquity, when it was proved that in some sense they are the “building blocks” of the s...

Kernel-Based Estimation of P(X less than Y)With Paired Data