The Singular Connection Between a Family of Exponential Functions and the Fermat-Wiles’ Theorem

1. Introduction

In 1637 Pierre Fermat formulated the conjecture known as his Last Theorem. In 1995, Andrew Wiles presented the first successful proof of Fermat's conjecture [1]. In this work, the semistable case of the Taniyama-Shimura theorem - previously a conjecture - which links modular forms and elliptic curves, was demonstrated. From this progress, combined with ideas from Frey and Ribet's theorem, the proof of the conjecture was obtained. This result was hailed as an astonishing breakthrough that opened the door to entirely new approaches to numerous problems and techniques [2]. The number of publications in recent years and the new links with other fields have confirmed this statement [3,4,5].

In parallel, research into exponential functions has evolved from the study of their basic properties towards deep problems connecting number theory, complex analysis, dynamical systems, and applied mathematics, keeping it at the frontier of mathematical knowledge: Schanuel's Conjecture, Dynamics of the Exponential Function, Transcendence of Hybrid Expressions and others [6,7]. Moreover, the exponential function is the natural language for describing a vast range of natural and human processes involving proportional change, making it a cornerstone of modern scientific modelling: Nuclear and Atomic Physics, Biology and Demography, Geology and Palaeoclimatology, Economics and Finance, Psychology and Neuroscience and many others [8]. It is therefore interesting to explore the possible links between Fermat - Wiles' theorem and exponential functions.

In this work, a singular connection between a family of exponential function and the Fermat – Wiles' theorem is revealed. This family depends on the variable x ∈ / x ≥ 0 and a parameter n ∈ Z⁺. The bases were chosen as integers, and an additional restriction was imposed on them. The functions have a unique maximum for each value of n, which is determined by an equation relating the bases. On the other hand, the sequence x_n, whose terms are the coordinates of the maxima for different values ​​of n, has a sup x_n. By analysing the limit of the sequence and the additional linking equation between the bases, a singular connection with the Fermat – Wiles' theorem is found.

Below, we proceed to carry out the analysis that reveals the aforementioned connection.

2. Analysis of the Connection

2.1. Family of Exponential Functions

Let the family of exponential functions (1) be described by the equation

$$\mathit{y}\left(\mathit{x},\mathit{n}\right)={\left(\mathit{n}\mathit{a}\right)}^{\mathit{x}}+{\left(\mathit{n}\mathit{b}\right)}^{\mathit{x}}-{\left(\mathit{n}\mathit{c}\right)}^{\mathit{x}}={\mathit{n}}^{\mathit{x}}\left({\mathit{a}}^{\mathit{x}}+{\mathit{b}}^{\mathit{x}}-{\mathit{c}}^{\mathit{x}}\right)$$

(1)

where a, b, c, n ∈ Z⁺ and

$$\mathit{c}>\mathit{b}>\mathit{a}$$

(2)

which are used as bases to define (1) in the interval x ⊂ [0, +∞). Additionally, the restriction (3) will be imposed on (1), where p ∈ Z⁺/ p ≥ 3

$$\mathit{y}\left(\mathit{p},\mathit{n}\right)=0$$

(3)

Next, we proceed to study the consequences of imposing restriction (3) on variation of function (1).

From (3) it can be simply shown that ∀ n and ∀ x ⊂ [0, p) holds that

$$\mathit{y}\left(\mathit{x},\mathit{n}\right)>0$$

(4)

On the basis of (2) and (4) it is easy to prove that

$$\mathit{y}\left(1,\mathit{n}\right)=\mathit{n}\left(\mathit{a}+\mathit{b}-\mathit{c}\right)\ge \mathit{n}$$

(5)

Also, from (3), applying the little theorem of Fermat (n^p ≡ n (mod p)) is possible to demonstrate (5).

Considering (3) and (5) as well the theorem of Intermediate Values for continuous functions, then ∃ $x_n^{*}$ ∈ [1, p] ⁄ $y\left(x_n^{*}, n\right)=1$. Therefore, at the endpoints of the interval [0, $x_n^{*}$], it holds that y(0, n) = y($x_n^{*}, n$) = 1. Consequently, by Rolle’s theorem for continuous and differentiable functions, there exists at least one point x = x_n / x_n ∈ [0, $x_n^{*}$] such that y_x′(x_n, n) = 0, and y(x_n, n) has an extremum at that value $(x_n < x_n^{*} <p)$.

From (1) it follows that

$${\displaystyle \frac{\mathit{d}\mathit{y}(\mathit{x},\mathit{n})}{\mathit{d}\mathit{x}}}=\left(\mathit{I}\mathit{n}\left(\mathit{n}\mathit{c}\right)\right)\left[{\left(\mathit{n}\mathit{a}\right)}^{\mathit{x}}\left({\displaystyle \frac{\mathit{I}\mathit{n}\left(\mathit{n}\mathit{a}\right)}{\mathit{I}\mathit{n}\left(\mathit{n}\mathit{c}\right)}}\right)+{\left(\mathit{n}\mathit{b}\right)}^{\mathit{x}}\left({\displaystyle \frac{\mathit{I}\mathit{n}\left(\mathit{n}\mathit{b}\right))}{\mathit{I}\mathit{n}\left(\mathit{n}\mathit{c}\right))}}\right)-{\left(\mathit{n}\mathit{c}\right)}^{\mathit{x}}\right]$$

(6)

$${\displaystyle \frac{{\mathit{d}}^2\mathit{y}(\mathit{x},\mathit{n})}{\mathit{d}{\mathit{x}}^2}}={\left(\mathit{I}\mathit{n}\left(\mathit{n}\mathit{c}\right)\right)}^2\left[{\left(\mathit{n}\mathit{a}\right)}^{\mathit{x}}{\left({\displaystyle \frac{\mathit{I}\mathit{n}\left(\mathit{n}\mathit{a}\right)}{\mathit{I}\mathit{n}\left(\mathit{n}\mathit{c}\right)}}\right)}^2+{\left(\mathit{n}\mathit{b}\right)}^{\mathit{x}}{\left({\displaystyle \frac{\mathit{I}\mathit{n}\left(\mathit{n}\mathit{b}\right)}{\mathit{I}\mathit{n}\left(\mathit{n}\mathit{c}\right)}}\right)}^2-{\left(\mathit{n}\mathit{c}\right)}^{\mathit{x}}\right]$$

(7)

lf y_x’(x, n) is evaluated at x = x_n, it is obtained that:

$${\left(\mathit{n}\mathit{a}\right)}^{{\mathit{x}}_{\mathit{n}}}{\displaystyle \frac{\left(\mathit{I}\mathit{n}\left(\mathit{n}\mathit{a}\right)\right)}{\left(\mathit{I}\mathit{n}\left(\mathit{n}\mathit{c}\right)\right)}}+{\left(\mathit{n}\mathit{b}\right)}^{{\mathit{x}}_{\mathit{n}}}{\displaystyle \frac{\left(\mathit{I}\mathit{n}(\mathit{n}\mathit{b}\right))}{\left(\mathit{I}\mathit{n}(\mathit{n}\mathit{c}\right))}}-{\left(\mathit{n}\mathit{c}\right)}^{{\mathit{x}}_{\mathit{n}}}=0$$

(8)

Relation (6) determines that y(x,n) has only one extremum. And if (2) and (8) are taken into account, it follows that y_x”(x_n, n) < 0, so the extremum at x = x_n is a maximum.

From y_x′(x_n, n) = 0, it results that

$${\left(\mathit{n}\mathit{a}\right)}^{{\left(\mathit{n}\mathit{a}\right)}^{{\mathit{x}}_{\mathit{n}}}}{\left(\mathit{n}\mathit{b}\right)}^{{\left(\mathit{n}\mathit{b}\right)}^{{\mathit{x}}_{\mathit{n}}}}={\left(\mathit{n}\mathit{c}\right)}^{{\left(\mathit{n}\mathit{c}\right)}^{{\mathit{x}}_{\mathit{n}}}}$$

(9)

Note that the coordinate of the maxima of the functions y(x,1), y(x,2),… and y(x, n)… are not equal.

In the next stage, the sequence consisting of the values ​​of the maxima of the family of functions (1) for different values ​​of n and the variation of the equality (9) as a function of n will be investigated.

2.2. Sequence of the Values of the Maxima

It will then be shown that the coordinates of the maxima of the family function y(x, n), x_n form a monotonically increasing sequence that has a limit.

The function (1) can be represented as the product of the functions f(x) and g(x)

$$\mathit{f}\left(\mathit{x}\right)={\mathit{n}}^{\mathit{x}}\mathit{g}\left(\mathit{x}\right)={\mathit{a}}^{\mathit{x}}+{\mathit{b}}^{\mathit{x}}-{\mathit{c}}^{\mathit{x}}$$

(10)

g(x) is defined at x ∈ [0, + ∞) and ∀ n ∃ only one maximum of y(x, n) at x = x_n. Also

$$\underset{\mathit{x}\to \mathit{p}}{\mathbf{lim}}\mathit{f}\left(\mathit{x}\right)\mathit{g}\left(\mathit{x}\right)=0$$

(11)

The derivative of y(x, n) is equal to

$${\displaystyle \frac{\mathit{d}\mathit{y}(\mathit{x},\mathit{n})}{\mathit{d}\mathit{x}}}={\mathit{n}}^{\mathit{x}}\left[\mathit{g}\left(\mathit{x}\right)+\mathit{g}\left(\mathit{x}\right)\mathit{I}\mathit{n}\left(\mathit{n}\right)\right]$$

(12)

Given n = k, at the coordinate x = x_k of the maximum of y(x, k) it is true that y_x′(x_k, k) = 0 and as n^k > 0 that implies that

$${{\mathit{g}}_{\mathit{x}}}^{\prime }\left({\mathit{x}}_{\mathit{k}}\right)=-\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)\mathit{I}\mathit{n}\left(\mathit{k}\right)$$

(13)

If y_x′(x, k+1) is evaluated at x = x_k, it results in

$${{\mathit{y}}_{\mathit{x}}}^{\prime }\left({\mathit{x}}_{\mathit{k}},\mathit{k}+1\right)={\left(\mathit{k}+1\right)}^{{\mathit{x}}_{\mathit{k}}}\left[\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)+\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)\mathit{I}\mathit{n}(\mathit{k}+1)\right]$$

(14)

The next step is to replace in (14) $\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)$ by its expression (13) obtaining

$${{\mathit{y}}_{\mathit{x}}}^{\prime }\left({\mathit{x}}_{\mathit{k}},\mathit{k}+1\right)={\left(\mathit{k}+1\right)}^{{\mathit{x}}_{\mathit{k}}}\left[-\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)\mathit{I}\mathit{n}\left(\mathit{k}\right)+\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)\mathit{I}\mathit{n}\left(\mathit{k}+1\right)\right]=={\left(\mathit{k}+1\right)}^{{\mathit{x}}_{\mathit{k}}}\mathit{g}\left({\mathit{x}}_{\mathit{k}}\right)\left(\mathit{I}\mathit{n}\left((\mathit{k}+1\right)-\mathit{I}\mathit{n}\left(\mathit{k}\right)\right)$$

(15)

Since k + 1 > k, In (k + 1) > In (k), ${\left(k+1\right)}^{x_k}>0$ and g(x_k) > 0, this means that ${{\mathit{y}}_{\mathit{x}}}^{\prime }\left(x_k,k+1\right)>0$. Therefore, y(x, k + 1) is growing at x = x_k. Considering (11) and ${{\mathit{y}}_{\mathit{x}}}^{\prime }\left(x_k,k+1\right)>0$, then the function y(x, k +1) must reach a maximum at some point x_k+1 > x_k. Therefore, the sequence formed by the coordinates of the maxima of x₁, x₂,…x_n… turns out to be monotonically increasing. Furthermore, considering that x_n < $x_n^{\mathrm{*}}$ < p, this is bounded by x = p

$${\mathit{x}}_1{\mathit{x}}_2\dots {\mathit{x}}_{\mathit{k}}{\mathit{x}}_{\mathit{k}+1}\dots \mathit{p}$$

(16)

This sequence is monotonically increasing and being bounded it converges to a supremum T according to the Monotonic Convergence theorem [9]

$$\mathit{T}=\mathbf{sup}{\mathit{x}}_{\mathit{n}}$$

(17)

So

$$\underset{\mathit{n}\to \mathit{\infty }}{\mathbf{lim}}{\mathit{x}}_{\mathit{n}}=\mathit{T}\le \mathit{p}$$

(18)

2.3. The Connection with the Fermat – Wiles' Theorem

In some cases, the use of the concept of limit allows to transcend the discrete and individual nature of integers to discern connections in the case of numerical functions [10]. This tool will now be used to determine the behaviour of (9) when n → ∞.

Performing simple algebraic transformations from (9) we obtain that

$${\mathit{a}}^{{\mathit{a}}^{{\mathit{x}}_{\mathit{n}}}}{\mathit{b}}^{{\mathit{b}}^{{\mathit{x}}_{\mathit{n}}}}{\mathit{c}}^{{-\mathit{c}}^{{\mathit{x}}_{\mathit{n}}}}={\mathit{n}}^{\left({\mathit{c}}^{{\mathit{x}}_{\mathit{n}}}-{\mathit{a}}^{{\mathit{x}}_{\mathit{n}}}-{\mathit{b}}^{{\mathit{x}}_{\mathit{n}}}\right)}$$

(19)

The behaviour of (19) when n → ∞ will be analysed below if T = p.

∀n y(0, n) > y(p, n) so p cannot be the coordinate of a maximum of y(x, n). If it is assumed that T = p and apply the limit n → ∞ to both sides of the equality (19), it is obtained

$${\mathit{a}}^{{\mathit{a}}^{\mathit{p}}}{\mathit{b}}^{{\mathit{b}}^{\mathit{p}}}{\mathit{c}}^{{-\mathit{c}}^{\mathit{p}}}={\left(\mathit{\infty }\right)}^0$$

(20)

the right side of the equality (20) is a fraction with a numerator and a denominator that are positive integers and its left side is an indeterminate form.

The performance of (19) when T < p is studied underneath.

From (2) and the hypothetical fulfilment of (3) it follows that

$${\mathit{a}}^{\mathit{T}}+{\mathit{b}}^{\mathit{T}}>{\displaystyle \frac{{\mathit{a}}^{\mathit{p}}}{{\mathit{c}}^{\mathit{p}-\mathit{T}}}}+{\displaystyle \frac{{\mathit{b}}^{\mathit{p}}}{{\mathit{c}}^{\mathit{p}-\mathit{T}}}}={\mathit{c}}^{\mathit{T}}$$

(22)

Then the exponent on the right side of (19) is negative. And if we calculate the limits of both sides of this equality when n → ∞ we obtain that

$${\mathit{a}}^{{\mathit{a}}^{\mathit{T}}}{\mathit{b}}^{{\mathit{b}}^{\mathit{T}}}{\mathit{c}}^{{-\mathit{c}}^{\mathit{T}}}=0$$

(23)

hence, a = 0 ∨ b = 0.

This analysis shows that if the family y(x, n) has a, b, c ∈ Z⁺ bases, then it cannot have roots x ∈ Z⁺/ x ≥ 3.

3. Conclusions

The previous analysis demonstrates a singular connection between a family of exponential functions and the Fermat – Wiles' theorem. If it is assumed that the bases of the family of exponential functions are integers and the existence of an entire root is imposed, then the convergence of the coordinates of the maxima and the equation determined by the fulfilment of the maximum condition leads to a contradiction. This analysis indicates that if the family y(x, n) has integer bases then it cannot have roots x ∈ Z⁺/ x ≥ 3.