So, to prove the Basic theorem of arithmetic we suppose that there exist equal natural numbers A, B consisting of different prime factors:

A=B (1)

where A=pp_>1p_>2 …p_>n; B=хx_>1x_>2 …x_>m ; n≥1; m≥1

Due to the equality of the numbers A, B each of them is divided into any of the prime numbers p_>i or x_>i. Each of the numbers A, B can consist of any set of prime factors including the same ones, but at the same time there is no one p_>i equal to x_>i among them, otherwise they would be in (1) reduced. Now (1) can be represented as:

pQ=xY (2)

where p, x are the minimal primes among p_>i, x_>i; Q=A/p; Y=B/x .

Since the factors p and x are different, we agree that p>x; x=p–δ_>1 then

pQ=(p – δ_>1)(Q+δ_>2) (3)

where δ_>1=p–x; δ_>2=Y–Q

From (3) it follows that Qδ_>1=(p – δ_>1)δ_>2 or

Qδ_>1=xδ_>2 (4)

Equation (4) is a direct consequence of assumption (1). The right side of this equation explicitly contains the prime factor x. However, on the left side of equation (4) the number δ_>1 cannot contain the factor x because δ_>1 = p – x is not divisible by x due to p is a prime. The number Q also does not contain the factor x because by our assumption it consists of factors p_>i among which there is not a single equal to x. Thus, there is a factor x on the right in equation (4), but not on the left. Nevertheless, there is no reason to argue that this is impossible because we initially assume the existence of equal numbers with different prime factors.

Then it remains only to admit that if there exist natural numbers A = B composed of different prime factors, then it is necessary that in this case there exist another natural number A_>1=Qδ_>1 and B_>1=xδ_>2; also equal to each other and made up of different prime factors. Given that δ_>1=(p–x)>2=(Y–Q)

A_>1 = B_>1, where A_>1>1

Now we get a situation similar to the one with numbers A, B only with smaller numbers A_>1, B_>1. Analyzing now (5) in the manner described above we will be forced to admit that there must exist numbers

A_>2=B_>2, where A_>2>1; B_>2

Following this path, we will inevitably come to the case when the existence of numbers A_>k=B_>k, where A_>k>k-1; B_>k>k-1 as a direct consequence of assumption (1) will become impossible. Therefore, our initial assumption (1) is also impossible and thus the theorem is proven.41

Looking at this very simple and even elementary proof by the descent method naturally a puzzling question arise, how could it happen that for many centuries science not only had not received this proof, but was completely ignorant that it had not any one in general? On the other hand, even being mistaken in this matter i.e. assuming that this theorem was proven by Euclid, how could science ignore it by using the "complex numbers" and thereby dooming itself to destruction from within? And finally, how can one explain that this very simple in essence theorem, on which the all science holds, is not taught at all in a secondary school?

As for the descent method, this proof is one of the simplest examples of its application, which is quite rare due to the wide universality of this method. More often, the application of the descent method requires a great strain of thought to bring a logical chain of reasoning under it. From this point of view, some other special examples of solving problems by this method can be instructive.

We will now consider another example of the problem from Fermat's letter-testament, which is formulated there as follows: