Is The Fundamental Theorem of Arithmetic flawed?

The long-admired and revered Fundamental Theorem of Arithmetic rests upon the definition of prime numbers. Essentially, it states that each whole number has a unique factorisation of primes (a bit like a ‘fingerprint’ or a chemical formula, e.g. H20—meaning 2 x hydrogen atoms and 1 x oxygen atom).

If we provide a simple application of the theorem it may help those readers unfamiliar with it. Take the random number 24 for example—factorising it we get, 24 = 2 x 12 = 2 x 2 x 6 = 2 x 2 x 2 x 3, and as this can be reduced no further into smaller prime factors, its unique factorisation-of-primes is as follows: 3 ‘atoms’ of the prime number “2”, and one ‘atom’ of the prime number “3”. [Remember that our definition of a prime number is that it can only be divided by itself and one.]

A pi in the ‘Fundamental’ ointment

Of the infinitude of primes “two” is the only even prime number. [Or, put another way, every prime number, except two, is odd.] If anything’s ‘odd’, that is! But many mathematicians hasten to add that, like the other primes, it is still only divisible by itself and one. Yet, to me, it smacks almost of an ‘inclusion of convenience’. Two must be included because half the integers are even and they must therefore have “2” as (at least) one of their factors. Also there is the fact that an odd times an odd number remains odd [since, (2n+1)(2m+1) = 4nm + 2n + 2m +1, and it is clear that this algebraic product is odd] . There’s an irrational pi in the ointment— within that circular jar of primes. “2” is the smallest, as well as the most common, of so-called primes in the Fundamental Theorem’s ‘decompositions into prime factors’. Since every alternate integer is even, the so-called prime number “2” must be a factor in the ‘prime factorisation’ of every alternate (i.e. even) integer. Whether the concept of ‘primality’ in number theory will be replaced with a better understanding of number relations and composition remains to be seen (I suspect it will). I merely raise the question (see below), and draw to the attention of open-minded readers some quite simple facts that point in new directions and to possibly fruitful areas of future research in number theory.

Interestingly, the Pythagoreans considered neither one nor two to be numbers at all— the first number to them was three since it possessed a beginning, middle, and end. (D Wells, 1997, pp.28-29).

I have since discovered much more than anecdotal support for the Pythagorean view. The set of Pythagorean Triples includes every integer (except one and two). This I proved for myself (from first principles and the well-known difference-of-squares identity) several years ago after I had read in a reputable book on Fermat’s Last Theorem that the Pythagorean Triples get rarer as you ascend the integers (supposedly like the primes). They do not. In fact they become more common (not rarer) as we ascend the integers, and moreover may be a member of more than one relatively-prime Pythagorean triple.

All of which raises a question mark and a new perspective on the supremacy currently occupied by the ‘primes’ in number theory (and thus the number “two” as a member of that set). Increasing exploration of this fascinating set of Pythagorean Triads and their interconnections may, in turn, one day completely overhaul our view of number relations, composition, and the ‘properties’ we ascribe to them (including shape and area). Moreover, we now have parametric forms of the Eutrigon and Co-eutrigon theorems which generate further interrelated triads for these new respective triangle classes in integral sides, and this opens up a whole new field of study to number theoreticians.

It does seem ironic, does it not, how in mathematics a single exception can dismiss a conjecture (that perhaps stood poised on the threshold of becoming inducted to the honour-roll of ‘theorem‘). And yet here we have exactly that— a single exception, an even prime, which has not even apparently raised any (or many) eyebrows, and is attached to a theorem so pivotal that it goes by the name of “Fundamental”.

Compound numbers

I suspect the ‘common-as-boots’ compound numbers will one day turn out to be more important than the primes. This seems in tune with the Principle of Universal Interconnectednessand to which I have alluded a number of times. It is intuitively ‘knowable’— beyond thought itself, and therefore beyond proof. Every new discovery about the Universe adds a little more to our awareness of the profound interconnectedness that pervades the Universe and of which we are part. Thus ipso facto, in the factorisation of numbers, ‘common factors’ are a form of connection which places compound numbers in a special class.

Consider the infinite number 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x11 x 12 x 13 x 14 … Such an infinite number is very interesting since it is the most rational of all numbers (other than one itself): every number is its factor, again reflecting the wholeness of the cosmos, and the miraculous way numbers may mirror that wholeness.

360° vs radians

Compound numbers are intrinsically more connected (by virtue of their factors). Some numbers like 360 go back three thousand years in history. But such a number was almost certainly no mere fluke of the base of system of numeration in vogue at the time (although it was in tune with it). The number “360” has a huge number of factors and, as it turns out,very important factors like 30, 60, 45, 90, 120, 180, and these no doubt facilitated the discovery of geometric theorems relating to triangles, squares, other polygons and circles*.

Carbon’s adaptability and valence

Carbon is well known for its ability to form various bonds with a host of other atoms (those requisite for life (hence the field known as organic chemistry). It can form: single bonds;, double bonds; alternating-single-and-double-bonds in the shape of a hexagon ring (benzene); geometric marvels of molecular organisation like buckminsterfullerene (a truncated icosahedron of consisting of 60 atoms); and more! This is certainly a ‘prime’ atom (in the popular sense of the word, ‘prime’) but stands diametrically opposed to the mathematical meaning of the word in its thoroughly ‘compound’ nature.

The sign of a number

In the traditional numberline view there exist positive and negative numbers extending in opposite directions from zero, and never meeting. In August 2005 I published a brief in which I demonstarted how the positive and negative integers could be ‘linked’ if the number plane is given a half-twist to form a Möebius band. This model has very interesting properties indeed and is in harmony with Quantum Mechanics in which observation is active in outcomes and ‘reading’.