Prime Numbers Part I The references

By Miguel Vidal Bravo - Jandia

1 Principle

I tell you: what do the times tables of 4, 16, and 21 mean to you in mathematics and physics? And in terms of prime numbers?

In Mathematics: the times table of 4: Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40... 16 is in this table (4 × 4 = 16), so it's a perfect square. So we see a hierarchy of powers of 2 here.

The times table of 21: Multiples of 21: 21, 42, 63, 84, 105... 21 = 3 × 7, so it's not a prime number. In physics:

4: Number often linked to dimensionality (4 dimensions in special relativity with spacetime).

16: Number of cells in a 4×4 square (linked to matrices and transformations in physics).

21: May be reminiscent of the 21 centimeters of the hydrogen line, used in astronomy to study the structure of the Universe.

Prime Numbers

4, 16, 21 are not prime.

4 = 2 × 2, 16 = 2^4, 21 = 3 × 7. But! 21 is the sum of the first four prime numbers: 2 + 3 + 5 + 7 = 21.

a/ Definition: A prime number is a natural number strictly greater than 1 that has only two divisors: 1 and itself. In other words, a number is prime if it cannot be written as the product of two natural numbers other than 1 and p.

Examples of prime numbers:

2 (the only even prime number), 3, 5, 7, 11, 13, 17, 19, 23, 29, etc.

Counterexamples (composite numbers):

4 = 2 × 2 → not prime

16 = 2 × 8 or 4 × 4 → not prime

21 = 3 × 7 → not prime

Curiosities about prime numbers: The smallest prime number is 2. There are an infinite number of prime numbers (proven by Euclid). They are fundamental because every natural number can be factored into a unique product of prime numbers (fundamental theorem of arithmetic). The distribution of prime numbers is a great mystery in mathematics (Riemann hypothesis, etc.).

Well, if I always tell you in mathematics 1 + 1 = 3, whereas in classical arithmetic, obviously 1 + 1 = 2.

So please remember that 1 + 1 = 3 and has never been 2, I'll prove this to you later. I'll save the concept for later. It's just like you were taught that the square is or is written as 2, so Einstein wrote E = MC2, which means energy is equal to mass and the speed of light squared, but look at what I just wrote: square. I rewrite: "square." A square has never been 2, but 4.

You see where I'm going with this: I'm challenging learned concepts, particularly the way mathematical formulas and notations are interpreted. On E = MC² and the notion of "square," there's an interesting criticism: I point out that in E = MC², the symbol "²" is called a "square" because it represents a multiplication by itself. But in geometry, a square is a 4-sided figure.

So, I draw a parallel between mathematical language and everyday language, and I question their validity. Indeed, the notion of "square" in mathematics isn't necessarily intuitive for everyone. We use the same word to refer to:

A power (²): in algebra, x2 means x multiplied by x, which is a misnomer, in my opinion. A geometric shape (a quadrilateral with 4 equal sides).

But where E=MC² is clear is that the notation C² does indeed mean "C multiplied by itself," not a square in the geometric sense. This is merely a historical misnomer that has led to "power 2" often being called a "square," some, if not all, would say.

b/ Discussion: I seek to show that classical mathematics is a human construct and that it is not absolute. Thus, if we use language abuses, how can we be sure that mathematics is a rigorous and reliable description of the world? Language abuse in mathematics is apparently common, and mathematics often uses terms that are not strictly defined as in everyday language.

Example 1: The word "square" is used to refer to x 2x2, whereas a square in geometry does indeed have four sides.

Example 2: We say that a line "is infinite," but it is never "physically" infinite; it is an abstraction.

Mathematicians create formal systems, but since they must explain these systems in human language, they inevitably make approximations and language abuses. Thus, if mathematicians don't express themselves very well, why trust them? As in law, qualification is extremely important; from this, the applicable system is derived. So, if someone doesn't express themselves very well, why assume they're reasoning well? Thus, an unsatisfactory qualification leads to a false deduction or even false reasoning.

Example: