This is nonzero when all prime factors of n of the form 4k + 3 appear with even exponents. … In 1973 it was proved that any even number can be written as the sum of a prime and a number that has at most two prime factors, not necessarily distinct. … Hardy and John Littlewood in 1922 to be proportional to 2n Apply[Times, Map[(# - 1)/(# - 2)&, Map[First, Rest[FactorInteger[n]]]]]/Log[n] 2 It was proved in 1937 by Ivan Vinogradov that any large odd integer can be expressed as a sum of three primes.

In the cases shown, the choice of whether to square or multiply by an additional factor of m at each step in computing m t is made on the basis of the successive digits in the base 2 representation of the number t .

That this must ultimately be the case one can see from the fact that the total number of elements in a substitution system can be multiplied by a factor from one step to the next, while in a cellular automaton the size of a pattern can only ever increase by a fixed amount at each step.

As mentioned on page 938 , through major mathematical effort, a complete classification of all finite so-called simple groups that in effect have no factors is known.

If k and n have no factors in common, there will be a t for which Mod[k t , n]  1 , so that the dot returns to position 1.

After the first step these systems give the same sequence of numbers, except for an overall factor of 3.

Each sequence of curves shows what happens when the local curvature is multiplied by a progressively larger factor.

But boxes (b) and (c) show that known alternative axiom systems for logic reduce the size of the axiom system by about a factor of ten.

Whenever natural selection is an important determining factor, I suspect that one will inevitably see many of the same simplifying features as in systems created through engineering.

But in fact there is a simple formula for the color of each square: given the largest factor in common between the