There appears to be some confusion with these terms among modern mathematicians. Sadly mathematics has become so removed from arithmetic by its application to the external world, that along with it goes a lack of clarity in thinking. Inevitable really. The further one gets from the roots of a thing, the more remote, the less do the fundamental rules appear. They become less apparent with distance from their source.
I think of myself as an Arithmetician and secondarily as a Geometer. Let’s look at a few of these definitions and consider their shortcomings.
Prime Numbers
A prime number is a whole number, greater than 1, that only has two factors. A factor is a number that divides another number evenly, with no remainder (https://www.bbc.co.uk › bitesize › articles)
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. (https://en.wikipedia.org/wiki/Prime_number)
The first definition does not take into account that any square of an odd number also has only two factors. The unit, 1, and the square root of the number under consideration. Consider 9 or 25 or 49. The factors of 9 are 1 and 3. These factors are shared with 3 itself, the originating Prime, commonly called the square root, of 9. However, 9 is not a Prime number.
Similarly with 25 where the originating prime is 5, which shares the factors 1 and 5 with the square root, 5. Again with 49 where the factors 1 and 7 are shared with the prime number 7. That all these squares have only a single root causes them to belong to a specific set of their own. This is defined as being incomposite when compared with another number. Taylor gives this as part of the definition of the species of odd number.
that number which is of itself second and composite,
but with reference to another first and incomposite.
And explains it further as
“The third species however, which is of itself second and composite, but when one number is compared to the other is first and incomposite, is obtained by the following method: The squares of the first and incomposite numbers, when compared to each other will be found to have no common measure.”
Taylor defines prime numbers as ‘First and incomposite’ which, he writes,
“is that which has no other part except that which is denominated from the whole quantity of the number; so that the part itself is no other than unity. And such are the numbers 3. 5. 7. 11. 13. 17. 19. 23. 29. 31.”
(from the Theoretic Arithmetic of the Pythagoreans, Thomas Taylor.)
However the fact that the factors are shared by the roots is not the point of this discussion. It is that the definition given by the BBC falls short of accuracy by claiming that a Prime is any number that has only two factors. Clearly this is not the case.
The definition in Wikipedia is hardly better.
What is clear is that Prime numbers form a Set within the natural series of numbers. Or if you prefer, Prime numbers are a Subset of the natural series of numbers.
If we accept that Prime numbers form a Set we have then to consider what is the nature of a set. Generally speaking a Set is composed of those things which share the properties of that Set.
If we consider the definition of a prime number to be ‘a number which shares two factors, the unit, 1 and the number itself’, as seems to be the case with modern mathematics, then we must consider the properties of that Set.
By this definition 2 may be included among the Prime numbers, as is generally the case today. It is divisible by 1, of which there are 2 within it, and by itself, 2, of which there is only 1. It is a condition of mutual reflection, if we so may describe it. The condition is found reflected in the other Prime numbers, in which there are the same number of units as the total of the number itself. So 5 is composed of five units of 1, or one unit of 5.
Let us next consider the nature and definitions of a Set.
“In mathematics, a set is a collection of different things; the things are elements or members of the set and are typically mathematical objects;” (https://en.wikipedia.org/wiki/Set_(mathematics))
If we accept this rather poor definition of a Set the crucial point, that they share something in common, i.e. the properties of the Set, is ignored.
Becoming intoxicated with the idea of Sets and Intersecting Sets, formulae have been developed that deal with these things. As with any algebraic formula no definition of the original expression is required, only to be defined at the final outcome of the formula. This removal from the original idea loses sight of the simplicity of the definition. To belong to a Set all things must share in the properties of that Set.
When we consider the Set of Prime numbers one thing becomes immediately apparent they are all Odd numbers. Except 2. Mathematicians get over this difficulty by declaring that 2 is the only even number that is also a Prime.
However the question arises if a thing is an exception to the general rule does it belong to the Set. To my mind it does not. A single exception within a larger body, all of which share the same properties as one another, is unacceptable since the many define the nature of the Set. The single exception simply does not belong to the Set defined by the others.
In this case the fuller definition of a Prime number is that ‘it is an Odd number which is divisible only by itself and the unit.’
A corollary to this is that in his extensive exploration, The Theoretic Arithmetic of the Pythagoreans, Thomas Taylor explains that the Ancients did not consider 1 and 2 to be numbers at all since, to belong to a Set it is necessary to display the properties of that Set. The only property that numbers hold in common is that a number when added to itself yields a sum which is less than the product of the number being multiplied by itself. This is clearly not the case with either 1 or 2. Each of these has peculiarities of its own. In the case of 2 the result of both addition and multiplication is the same, 4. This is unique. Similarly with 1 the result is the direct inverse of that of numbers, in that addition yields a sum which is greater in quantity than the result of multiplying 1 by itself.
In this way 2 may be seen to offer a plane surface through which the ideal contained in The One reflects into material expression wherein the images of the ‘real’ are inverted, in comparison with the ideal version held in The One. This principle of reflection is fundamental to creation and is demonstrated in various ways, as well as elsewhere in Arithmetic.