We begin our study with the formulation of the FGT from Fermat's letter to Mersenne in 1636. It is presented there as follows:

Every number is equal

one, two or three triangles,

one, 2, 3 or 4 squares,

one, 2, 3, 4 or 5 pentagons,

one, 2, 3, 4, 5 or 6 hexagons,

one, 2, 3, 4, 5, 6 or 7 heptagons,

and so on to infinity [36].

Since polygonal numbers are clearly not respected by today's science, we will give here all the necessary explanations. The formula for calculating any polygonal number is represented as

m_>i = i+(k−2)(i−1)i/2

where m is a polygonal number, i is a serial number, k is the quantity of angles.

Thus, m_>1=1; m_>2=k; and for all other i the meaning of mi varies widely as shown in the following table:

Table 1. Polygonal numbers

To calculate m_>i it is enough to obtain only triangular numbers by the formula, which is very easily since the difference between them grows by unit with each step. And all other m_>i can be calculated by adding the previous triangular number in the columns. For example, in column i=2, numbers increase by one, in column i=3 – by three, in column i=4 – by six etc. i.e. just on the value of the triangular number from the previous column.

To make sure that any natural number is represented by the sum of no more than k k-angle numbers is quite easily. For example, the triangular number 10 consists of one summand. Further 11=10+1, 12=6+6, 13=10+3 of two, 14=10+3+1 of three, 15 again of one summand. And so, it will happen regularly with all natural numbers. Surprisingly that the number of necessary summands is limited precisely by the number k. So, what is this miraculous power that invariably gives such a result?

As an example, we take a natural number 41. If as the summand triangular number will be closest to it 36, then it will not in any way to fit into three polygonal numbers since it consists minimum of 4 ones i.e. 41=36+3+1+1. However, if instead of 36 we take other triangular numbers for example, 41=28+10+3, or 41=21+10+10 then again in some unknown miraculous way everything will so as it stated in the FGT.

At first glance it seems simply unbelievable that it can somehow be explained? But we still pay attention to the existence of specific natural numbers, which are consisting at least of k k-angle numbers and denoted by us as S-numbers. Such numbers are easily to find for example, for triangles – 5, 8, 14, for squares – 7, 15, 23, for pentagons – 9, 16, 31 etc. And this our simple observation allows us directly to move to aim i.e. without using ingenious tricks or powerful "sharpness of mind".

Now to prove the FGT, suppose the opposite i.e. that there exists a certain minimal positive integer N consisting minimum of k + 1 k-angle numbers. Then it’s clear that this our supposed number should be between some k-angle numbers m_>i and m_>i+1 and can be represented as

N=m_>i+δ_>1 where δ_>1=N−m_>i (1)

It is quite obvious that δ_>1 must be an S-number since otherwise this would contradict our assumption about the number N. Then we proceed the same way as in our example with the number 41 i.e. represent the supposed number as

N = m_>i-1+δ_>2 where δ_>2=N−m_>i-1

Now δ_>2 should also be an S-number. And here so we will go down to the very end i.e. before

δ_>i-1=N−m_>2 =N−k and δ_>i=N−m_>1=N–1 (2)

Thus, in a sequence of numbers from δ_>1 to δ_>i, all of them must be S-numbers i.e. each of them will consist of a sum minimum of k k-angle numbers, while our supposed number N will consist minimum of k+1 k-angle numbers. From (1) and (2) it follows:

N− m_>i =S_>i (3)

Thus, if we subtract any smaller polygonal number m