I.4: More Determining Whether Different Mathematical Properties are Compatible, Working With Arguments that are not Fully Rigorous, Conditional Results, Numerical Evidence, Illegal Calculations, Finding Explicit Proofs and Algorithms, Pages 67 – 72.

A group, G, is called finitely generated if there is a finite subset of elements of G, say X, such that the elements in G that are not in X can be written as products of the elements in X.  The set of all 2 by 2 matrices of determinant one and entries that are integers is called SL2(Z).  SL2(Z) is finitely generated.

An element, x, of a group, G, is said to be of finite order if there is some power of x that equals the identity of the group G.  The smallest such power is called the order of x.

In a group with order of all elements, except the identity, equal to 2 the group is Abelian, that is, ab = ba.  This shows that the general property of finite order equal to 2 implies the Abelian property.

A finite group G has a minimal set of generators if every element of the group can be formed from the generators and all of the generators are required to do this.

For the group, G, with order 2 and a minimal set of generators one can define a standard form.  This is done by ordering the elements the same was as the generators are ordered and by eliminating or reducing and powers greater than 1 of any multiply occurring generator.  Thus G has at most 2k elements.  This means that G is finite.

A finitely generated group of order 2  implies that G is finite.

The Burnside problem is: if G is finitely generated of order n then is G finite?  Yes, for n = 3.  It was shown by Ivanov in 1992 that G does not have to be finite for n ≥ 13.  It is still not known whether a group with 2 generators of order 5 must be finite.

There are arguments that are not fully mathematically rigorous.  These include conditional results, numerical evidence and illegal calculations.

The Riemann hypothesis is the most famous unsolved problem in mathematics.  The Riemann hypothesis is equivalent to the statement that p(n) and li(n) differ by at most c √nlog(n) for some constant c.  If true this would mean li(n) is a good approximation to p(n).  Here li(n) =  Integral(0,N) 1/log(t) dt.  Although unproven many arguments depend upon the Riemann hypothesis. 

 Another unproven problem is whether NP problems are equivalent to P problems.    

 Algebraic number theory has many conjectural statements.

The Goldbach conjecture states that every even number greater than or equal to 4 is the sum of two primes.  There is numerical evidence for this as it has been checked for ever even number up to 1014.  There is an even stronger version of Goldbacj’s conjecture.

It is sometimes the case that nonrigorous calculations lead to conjectures that may or may not be true.  Sometimes physicists provide such arguments that appear to lead to true conjectures.  These may include having something tend to zero that doesn’t make sense at zero or having a limiting form of integers rather than continuously defined real numbers.

There may be proofs that show existence of some object but give no clue as to how to find such an object.  The real numbers are dominated by transcendental numbers but only a few of them are known explicitly, for example, p.

One can sometimes find clever ideas to overcome difficulties.  An example is provided of finding volumes of convex objects in high dimensions.

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