Application of diophantine equations for practical problems solution in biology

For this reason, in particular, an analysis was made and an article was published, [2], in which it was shown that equations with square roots and roots with any degrees from differences can be broken/decomposed up into different terms. This is equation (143) in this article. It was impossible to neglect the attempt to connect the obtained results with the verifi cation of the Fermat’s theorem. Mathematical modeling in biotechnology, [2 and literature in this article] changed its course since last 3–4 decades with a shift toward situation-specifi c, complex treatments as opposed to causal, mechanistic and general analyses as in the past. Common patterns/theories of a cell population growth do not receive much emphasis unlike in the past, specifi cally the well-known variations from Monod’s model. On the other hand, development of fermentation technology focused only on improving fermenter technology and automation. Thus, the mathematical and physical (biological) problems of microbial growth and biosynthesis, both in the general case as well as in specifi c applications, remained unattended too.


Introduction
The Fermat's theorem is one of the most popular theorems in mathematics. Its condition is formulated simply at the school arithmetic level. However, many mathematicians and laymen have been looking for the proof of the theorem for more than three hundred years. This theorem says that at the same time there are no nonzero integers "a", "b", "c", "n", for n > 2, which would satisfy the equation: a n +b n =c n , (1).
It was proved in 1994 by A. Wiles and colleagues, [1]. The proposed proof is very good for understanding in a narrow mathematical community. However, we can say that not all specialists in mathematics understand this. Simplifi cation is necessary for a wider range of stakeholders.
For this reason, in particular, an analysis was made and an article was published, [2], in which it was shown that equations with square roots and roots with any degrees from differences can be broken/decomposed up into different terms. This is equation (143)  The author of this article published a preprint, [3], with his own proof of FLT, which at the time of the writing has not yet received serious objections to the presented proof of FLT, despite the fairly wide distribution of this preprint among the professional community of "pure mathematics". a = (c n -b n ) (1/n) = (ε/γ) (1/n) [(c n ) (1/n) Φ (n-1)/n -φ (n-1)/n ((b n )/γ) 1/n ],

Background information
Naturally, the inverse calculation of "b" through "a" is also possible.
2.1.2 «a», «b», «c», «n» are integers for the case considered in this article, and a = pA, b = pB, c = pC, where "p" is any other integer. The Pythagorean primitive triples "A", "B", "C" are the minimum integers greater than 1 that satisfy to the condition (1), as well as the numbers "a", "b" and "c", moreover, a < b and c > b; 2.1.3 C 0 =C n / 2 is the ratio of C n to the number Φ 2 , where and γ B = B n /C 0 (4) Here the indices "A" or "B" with the numbers "" indicate through which parameter this "" is calculated. Accordingly, for the " A " value, the calculation was carried out through the parameter "A" for the further purpose of the "B" value study and vice versa. Naturally, it is possible to use «a», «b», «c» to determine  a and  b ; The values obtained in accordance with paragraph 2.  i.e., L A =( n - n-2  A ) 1/n C, (6), and L B =( n - n-2  B ) 1/n C, (7).

Solution of the Equation (2) for different n
The Equations (6) and (7) were obtained at simplifying of the fi rst term in the right-hand side of equation (2). Calculations are made through "A" or "B" upon receipt of values corresponding to "B" or "A". That is, the Equation (3) is used in the analysis of "B" values, and the Equation (4) is used in the analysis of "A" values. Accordingly, the values with the corresponding indices "L A " and/or "L B " in the form of the Equations (6) and (7) were obtained after simplifi cation.
The Equations for the quantities "M A " and "M B " were also obtained similarly to the Equations (6) and (7): , .

2.2.2
The joint solution of the Equations (6), (8) and (7), (9) by equating the factors " A " and " B " in each of the corresponding equations allows us to write the following expressions: , . (10) and (11) can be represented as follows:

Practical Application and Evidence in the Nature
The Equation (18) [2]. The biological signifi cance of this phenomenon can only be explained by the fact that there is no sharp boundary between the properties of X st and X div cells, which are close in the age of its cell cycles. This refers to the situation if 2 daughter cells have just come from one parent cell and one of them has entered into the X div state through some small time interval compared to the doubling time, t d , and the second has remained in the X st state.
But, it is obvious, the Equations (12) and (13) can also have integer solutions according to paragraph 2. This is acceptable for the case if we assume that C = X 1/2 and B = (X new ) 1/2 , where the X new values should be considered as "new" cells, [2], which were formed after some fi xed reference time. For example, after a time corresponding to the time for the boundary between the exponential growth phase and the slow growth phase,  Lim .  Accordingly, cells, which were already existing in a population at the given time,  Lim , are "old" cells, X old , [2]. Moreover, A = (X old ) 1/2 , where A values is also an integer, like «C» and «B» values. At  =  Final , both components, X new or X old , cannot be equal to 0.
It's obvious that: Thus, the equality is true:  (1) and (2)). The Equations (26) and (27) were fi rstly shown in [4], however, with an incorrect interpretation like X div and X st , respectively.
Taking into account all the equations of this article and article [2], it was shown that the development of populations can be determined by the Fermat's theorem and the Pythagorean Theorem, as a special case of Fermat's theorem. The reason is: there are no other possibilities for the Nature, except n = 2.

Conclusion
The Fermat's theorem and Pythagorean Theorem can be used by a relatively simple algebraic way to analyze S-shaped growth curves of microorganisms.
Application in practical biology for describing of population growth was proposed to show essential features in interpretations of belonging to different groups of cells.