Pierre de Fermat

Pierre de Fermat's mathematical and physical architecture is constructed upon four core paradigms:

• Method of Adequality (Ad-aequalitas): A foundational algebraic precursor to differential calculus, determining extrema and tangents by comparing a function $f(x)$ with $f(x+e)$, setting them approximately equal, dividing by increment $e$, and evaluating at $e=0$.
• Fermat's Principle of Least Time: Light traveling between two points follows the path that minimizes total propagation time $\int dt$, rather than spatial distance.
• Foundations of Diophantine Number Theory: Systematic investigation of integer properties, prime distribution, and modular arithmetic, establishing modern number theory.
• The Method of Infinite Descent (Descente Infinie): A rigorous proof technique demonstrating the non-existence of integer solutions by showing that any purported solution implies an infinitely descending sequence of strictly smaller positive integers, which is impossible in the well-ordered natural numbers $\mathbb{N}$.

+--------------------------------------------------------------+
|                FERMAT'S VARIATIONAL ARCHITECTURE             |
|                                                              |
|        Optical Path Optimization:     Geometric Extrema:     |
|        Fermat's Principle             Method of Adequality   |
|        T = \int dt = minimum          f(x + e) \approx f(x)  |
|                 \                             /              |
|                  v                           v               |
|            [ VARIATIONAL PRINCIPLE: \delta S = 0 ]           |
|            - Shortest optical time (Snell's Law)             |
|            - Horizontal tangent / extremum condition         |
|                               |                              |
|                               v                              |
|            [ DISCRETE NUMBER THEORY: DIOPHANTUS ]            |
|            - Fermat's Little Theorem: a^p \equiv a (mod p)   |
|            - Infinite Descent: x^4 + y^4 = z^2 has no int sol|
|            - Fermat's Last Theorem: x^n + y^n \ne z^n (n > 2)|
+--------------------------------------------------------------+

2.1 The Method of Adequality for Extrema (1636)

To find the maximum or minimum of a continuous polynomial expression $f(x)$:

1. Substitute $x + e$ into the expression: $f(x + e)$.
2. Set the two expressions adequal (approximately equal): $$f(x + e) \ \approx \ f(x)$$
3. Expand algebraically and subtract $f(x)$ from both sides: $$f(x + e) - f(x) \ \approx \ 0$$
4. Divide all terms by the non-zero increment $e$: $$\frac{f(x + e) - f(x)}{e} \ \approx \ 0$$
5. Delete all remaining terms containing $e$ (equivalent to setting $e = 0$): $$\left[ \frac{f(x + e) - f(x)}{e} \right]_{e=0} = 0$$

Example: Maximize the area of a rectangle with perimeter $2b$: $A(x) = x(b - x) = bx - x^2$. $$b(x + e) - (x + e)^2 \approx bx - x^2$$ $$bx + be - x^2 - 2xe - e^2 \approx bx - x^2$$ $$be - 2xe - e^2 \approx 0 \implies b - 2x - e \approx 0$$ Setting $e = 0$: $$b - 2x = 0 \implies x = \frac{b}{2} \quad (\text{a square minimizes perimeter/maximizes area})$$

2.2 Derivation of Snell's Law from Principle of Least Time (1662)

Let light travel from point $A(0, a)$ in medium 1 (velocity $v_1$) to point $B(c, -b)$ in medium 2 (velocity $v_2$), crossing the horizontal interface at $(x, 0)$.

Total transit time $T(x)$: $$T(x) = \frac{\sqrt{x^2 + a^2}}{v_1} + \frac{\sqrt{(c - x)^2 + b^2}}{v_2}$$

Applying the extremum condition via adequality ($dT/dx = 0$): $$\frac{d T}{dx} = \frac{x}{v_1 \sqrt{x^2 + a^2}} - \frac{c - x}{v_2 \sqrt{(c - x)^2 + b^2}} = 0$$

Recognizing the geometric sine of the angles of incidence $\theta_1$ and refraction $\theta_2$: $$\sin\theta_1 = \frac{x}{\sqrt{x^2 + a^2}}, \quad \sin\theta_2 = \frac{c - x}{\sqrt{(c - x)^2 + b^2}}$$

Yielding Snell's Law: $$\frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2} \implies n_1 \sin\theta_1 = n_2 \sin\theta_2$$ where refractive index is defined as $n \equiv c_0 / v$.

2.3 Fermat's Little Theorem (1640)

If $p$ is a prime number and $a$ is an integer not divisible by $p$ ($\gcd(a, p) = 1$): $$a^{p-1} \equiv 1 \pmod{p}$$ Equivalently, for any integer $a$: $$a^p \equiv a \pmod{p}$$

Proof via residue classes: Consider the set of non-zero residues modulo $p$: $S = \{1, 2, \dots, p-1\}$. Multiply each element by $a$: $a S = \{a \cdot 1, a \cdot 2, \dots, a \cdot (p-1)\}$. Because $\gcd(a, p) = 1$, the elements of $a S$ form a permutation of $S$ modulo $p$. $$\prod_{k=1}^{p-1} (a \cdot k) \equiv \prod_{k=1}^{p-1} k \pmod{p}$$ $$a^{p-1} (p-1)! \equiv (p-1)! \pmod{p}$$ Since $\gcd((p-1)!, p) = 1$, we divide by $(p-1)!$: $$a^{p-1} \equiv 1 \pmod{p}$$

3.1 Geometric Schematic of Fermat's Principle of Least Time

            Medium 1 (Air, Speed v1)
            Point A (0, a)
                 \
                  \  Incident Ray: Distance d1 = \sqrt(x^2 + a^2)
                   \ Angle of incidence \theta_1
                    \ |
     ----------------+--------------------------- Interface
                    (x, 0) | Angle of refraction \theta_2
                           \
                            \ Refracted Ray: Distance d2 = \sqrt((c-x)^2 + b^2)
                             \
                        Point B (c, -b)
            Medium 2 (Water, Speed v2 < v1)

Principle: Path chosen minimizes Total Time T = d1/v1 + d2/v2 !

3.2 The Infinite Descent Proof Topology ($x^4 + y^4 = z^2$)

   Assume Integer Solution (x0, y0, z0) with minimal positive z0:
                         |
                         v
   Apply Pythagorean Triple Parameterization:
   x0^2 = 2uv,  y0^2 = u^2 - v^2,  z0 = u^2 + v^2
                         |
                         v
   Algebraic Reorganization yields a NEW integer solution:
   (x1, y1, z1) satisfying x1^4 + y1^4 = z1^2
                         |
                         v
   Rigorous Inequality: 0 < z1 <= z1^2 < z0
                         |
                         v
   CONTRADICTION!
   No strictly descending sequence of positive integers can exist in N!
   Therefore, no integer solution can exist.

• Verification of Snell's Law (1662): Fermat derived the exact empirical law observed by Willebrord Snell (1621) and René Descartes (1637) from pure variational kinematics.
• Universal Cornerstone of Cryptography: Fermat's Little Theorem ($a^{p-1} \equiv 1 \pmod p$) forms the mathematical basis for the RSA public-key cryptosystem (Rivest, Shamir, Adleman, 1977) and Miller-Rabin primality testing.
• Formal Proof of Fermat's Last Theorem (Andrew Wiles, 1994): Proved the modularity theorem for semistable elliptic curves, establishing that $x^n + y^n = z^n$ has no non-zero integer solutions for $n > 2$.
• Sum of Two Squares Theorem: Fermat proved that an odd prime $p$ can be written as a sum of two squares ($p = x^2 + y^2$) if and only if $p \equiv 1 \pmod 4$ (e.g. $5 = 1^2 + 2^2$, $13 = 2^2 + 3^2$).

5.1 The Fermat Primes Conjecture Failure

Fermat conjectured that all numbers of the form $F_n = 2^{2^n} + 1$ are prime. - $F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257, F_4 = 65,537$ (all prime). In 1732, Leonhard Euler disproved Fermat's conjecture by showing that $F_5$ is composite: $$F_5 = 2^{32} + 1 = 4,294,967,297 = 641 \times 6,700,417$$ To date, no other Fermat primes beyond $F_4$ have ever been found.

5.2 The 'Marvelous Proof' Margin Anachronism

Modern consensus among mathematicians and historians (Edwards, Ribenboim, Wiles) is that Fermat did not possess a valid general proof for his Last Theorem. The algebraic machinery of modular forms, Galois representations, and Iwasawa theory required to prove the theorem did not exist in the seventeenth century; Fermat likely devised a flawed induction based on infinite descent that works only for $n = 4$.

5.3 Ambiguity in the Formal Limit Concept

Fermat's adequality lacked a formal $\epsilon$-$\delta$ limit definition. Setting increment $e$ to zero after dividing by $e$ caused conceptual discomfort among contemporaries, who accused Fermat of dividing by zero, an ambiguity that persisted until Augustin-Louis Cauchy formalized limits in 1821.

• Methodus ad disquirendam maximam et minimam et de nequis, 1636 (circulated manuscript; first published in Varia Opera, 1679).
• De aequationum localium transmutatione et emendatione, 1637.
• Analyse pour les réfractions, 1662 (introduction of Fermat's Principle).
• Observations sur Diophante, in Diophanti Alexandrini Arithmeticorum libri sex (edited by Samuel de Fermat), Bosc, Toulouse, 1670 (contains marginal note of Last Theorem).
• Varia Opera Mathematica D. Petri de Fermat, Joannis Pech, Toulouse, 1679.
• Wiles, A., Modular Elliptic Curves and Fermat's Last Theorem, Annals of Mathematics, Vol. 141, No. 3, pp. 443–551, 1995.