Welcome to the javaphysics.com

Home

Friday, 20 November 2009 00:49

A number is called "normal" with respect to a given base if, when the number is expressed in that base, the asymptotic frequencies of occurrence of each distinct string of k digits are equal, and this applies to every positive integer k. For example, if a number is normal in the base 10, the asymptotic frequency of occurrence of each of the decimal numerals 0, 1, …, 9 is precisely 1/10, and the asymptotic frequency of each two-digit strings 00, 01, 02, ..., 99 is exactly 1/100, and so on. By the same token, the asymptotic frequency of occurrence of each 10-digit string must be exactly 1/10¹⁰. Now, of the distinct 10-digit strings, exactly 10! contain each numeral just once, i.e., there are 10! permutations of the 10 decimal numerals 0, 1, ..., 9. Therefore, if a number is normal in the base 10, the asymptotic frequency of 10-digit strings comprising permutations of all 10 numerals must be 10!/10¹⁰, which equals roughly 1/2755.73.

Symmetric Pseudoprimes

Friday, 20 November 2009 00:45

Fermat's Little Theorem states that for any integer c, if p is a prime, then c^p - c is divisible by p. This is a necessary but not quite sufficient condition for primality, because there are (rare) composites that satisfy this type of test for certain values of c. For example, 2³⁴¹ - 2 is divisible by 341, even though 341 is composite. This is why 341 is called a pseudoprime relative to the base 2. There are even composites, called Carmichael numbers, that are pseudoprimes relative to every integer base. The smallest Carmichael number is 561.

Geometry of the Universe

Thursday, 19 November 2009 11:10

general relativity allows for spacetime to be curved, thus the whole Universe may have a non-flat geometry
three possible shapes are allowed, flat, positive or negative curvature

Can the Universe be finite in size? If so, what is ``outside'' the Universe? The answer to both these questions involves a discussion of the intrinsic geometry of the Universe.

There are basically three possible shapes to the Universe; a flat Universe (Euclidean or zero curvature), a spherical Universe (positive curvature) or a hyperbolic Universe (negative curvature). Note that this curvature is similar to spacetime curvature due to stellar masses except that the entire mass of the Universe determines the curvature. So a high mass Universe can have positive curvature, a low mass Universe might have negative curvature.