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It is known that any odd perfect number must be greater than 10 300 , must have a factor of at least 10 6 , and must be less than 4 4 s if it has only s prime factors.
In general the pattern of probabilities for changes can be thought of as being somewhat like a Green's function in mathematical physics—though the nonadditivity of most cellular automata makes this analogy less useful.
There is also exchange of DNA between paternal and maternal chromosomes, typically with a few crossovers per chromosome, at positions that seem more or less randomly distributed among many possibilities (the details affect regions of repeating DNA used for example in DNA fingerprinting).
Ductile materials in effect flow slightly before breaking, and as a result their fracture surfaces tend to be less jagged.
Less well recognized is the fact that formulating mathematical ideas in a Mathematica program is at least as effective a way to produce clarity of thinking and understanding as formulating a traditional proof.
An important observation, related to the fact that limitations in color ranges are usually not too troublesome, is that the perceived colors of objects stay more or less constant even when viewed in very different lighting, corresponding to very different wavelength distributions.
In the 1980s, at least in some fields, traditional statistical analysis began to become less popular, being replaced by more direct examination of data presented graphically by computer.
Multicolor Turing machines [from 2-color TMs]
Given rules in the form on page 888 for a Turing machine with s states and k colors the following yields an equivalent Turing machine with With[{c = Ceiling[Log[2, k]]}, (3 2 c + 2c - 7) s] states (always less than 6.03 k s ) and 2 colors:
TMToTM2[rule_, s_, k_] := # /.
In recent years, however, the increasing use of explicit computation has made proof less important, at least in most applications of mathematics.
This is the simplest polynomial giving Fibonacci[n] , and there are for example no polynomials with 2 variables, up to 4 terms, total degree less than 4, and integer coefficients between -2 and +2, that give any of 2 n , 3 n or Prime[n] .