Abstract

Fermat’s Last Theorem — the assertion that there are no nonzero integer solutions to the equation

x^n + y^n = z^n

for n > 2 — remained unproven for 358 years. Pierre de Fermat’s marginal note in his copy of Diophantus’ Arithmetica claimed he had “a truly marvelous proof” too large to fit in the margin. This white paper examines the mathematical techniques known to Fermat, the context of his other work, and plausible reconstructions of how he might have believed he had proven the theorem. We will explore:

The mathematical tools available to Fermat in the mid-17th century. Fermat’s own favored proof strategies. Known partial proofs he provided for special cases. Possible misunderstandings or overextensions of correct methods. Hypotheses from historians and mathematicians about what his “marvelous proof” could have been.

1. Historical and Mathematical Context

1.1. Mathematical Landscape in Fermat’s Time

In the 1630s–1660s, European number theory was in its infancy. Algebraic notation was becoming standardized, but concepts like rings, fields, and modular arithmetic had not yet been formalized. Fermat corresponded with contemporaries such as Mersenne, Pascal, and Huygens, but there was no professional mathematical community in the modern sense.

Available to Fermat:

Euclid’s Elements (Greek geometry and number theory). Diophantus’ Arithmetica (Greek number theory, mainly on rational solutions to equations). The method of infinite descent (developed by Fermat). Properties of Pythagorean triples and sums of squares. Early algebraic manipulations influenced by Viète and Descartes.

Not available to Fermat:

Group theory, field theory, or algebraic number theory. Modular arithmetic in its modern form (though he knew specific modular congruences). Elliptic curves and Galois representations.

2. Fermat’s Preferred Proof Techniques

2.1 Infinite Descent

Fermat’s most characteristic approach was to show that if a solution to a Diophantine equation existed, a smaller one could be found, leading to an infinite regress — impossible for positive integers. He applied this to:

n = 4 case of the Last Theorem. Some problems about sums of two squares.

Key idea: Prove impossibility by contradiction via a descending sequence of solutions.

2.2 Transformation to Known Problems

Fermat often tried to relate a new equation to one he already had solved, via algebraic manipulation, factorization, or geometric representation.

2.3 Factorization in Special Rings

While Fermat did not have full algebraic number theory, he experimented with factorization in special cases, like the Gaussian integers \mathbb{Z}[i] for sum-of-two-squares results.

3. Fermat’s Known Work on the Last Theorem

Fermat did produce a valid proof for the n = 4 case:

x^4 + y^4 = z^4

by showing it reduces to x^4 + y^4 = z^2, then using infinite descent to show impossibility.

He also claimed to have proven n = 3, but no valid proof survives — and historians suspect he made a subtle error.

Since any exponent n > 2 can be reduced to the cases n = 3 or n = 4 via factorization, Fermat may have believed that solving both cases would solve all.

4. Possible Routes Fermat Might Have Believed Worked

4.1 Reduction to Squares (n=4 Case) and Generalization

Fermat may have thought:

For even n: reduce to n = 4 using x^{2k} + y^{2k} = z^{2k} ⇒ (x^k)^2 + (y^k)^2 = (z^k)^2 style manipulations. For odd n: reduce to n = 3, then apply a similar infinite descent argument.

This is half right — the n = 4 proof works, but n = 3 proof is nontrivial and Fermat’s attempts likely failed.

4.2 Misapplication of Infinite Descent to Odd Exponents

The descent for n = 4 relies on special factorization over integers:

x^4 + y^4 = (x^2 + y^2 + \sqrt{2}xy)(x^2 + y^2 – \sqrt{2}xy)

and geometric parametrization of Pythagorean triples. Fermat might have assumed a similar approach works for cubes or higher odd powers — but without the right algebraic number theory, it doesn’t close.

4.3 Overconfidence from Success with Sums of Squares

Fermat’s triumph with the theorem:

p \equiv 1 \ (\text{mod } 4) \implies p = a^2 + b^2

and related results might have convinced him that all Diophantine problems of this flavor were tractable with similar descent methods.

4.4 Possible Factorization in Complex Integers

Although not formalized, Fermat may have tried to work with integers of the form a + b\omega where \omega is a root of unity, mirroring modern proofs for n=3 and other small exponents.

He could have stumbled upon partial factorizations like:

x^3 + y^3 = (x + y)(x^2 – xy + y^2)

and thought they generalized cleanly.

5. Why Fermat Was Likely Wrong About Having a Complete Proof

The n = 4 proof works — but is not trivially generalizable. The descent argument for odd n would require tools not yet developed. Later mathematicians (Euler, Sophie Germain, Kummer) had to invent entirely new theories to handle specific exponents, showing the depth of the problem. Fermat often worked quickly and overestimated the generality of his methods — some of his other “proofs” in correspondence contain small but critical gaps.

6. Modern Historians’ Reconstructions

Leading hypotheses:

Sophie Germain’s Hypothesis: Fermat may have stumbled upon a primitive form of her theorem for certain primes, believed it held generally, and didn’t notice the exceptions. n = 4 + n = 3 Reduction Hypothesis: He thought proving n = 3 and n = 4 sufficed, and believed he had both. Overgeneralized Descent Hypothesis: He mistook a structural similarity in descent arguments for a proof of the general case.

7. Conclusion

Fermat’s “marvelous proof” likely consisted of:

A valid infinite descent proof for n = 4. A mistaken or incomplete attempt to extend this proof to odd exponents (probably starting with n = 3). Overconfidence from other successes in number theory.

His confidence was not unusual for mathematicians of his era, but the fact that his claim stood unverified for centuries is a testament to how deceptively deep the problem is. In the end, it required the machinery of modern algebraic geometry, elliptic curves, and modular forms — developments wholly beyond Fermat’s horizon.