Abstract
For $n\ge3$, $x^n+y^n=z^n$ has no solution in coprime positive integers (Fermat’s Last Theorem, proved by Wiles in $1995$). This paper proves nothing; it sets out the roads that reach this “no solution” and counts which exponents each reaches. No new theorem or law is claimed.
First, the road of infinite descent reaches $n=4$——$x^4+y^4=z^2$ has no solution up to $3000$, while the same search found $534$ coprime solutions of $x^2+y^2=z^2$. If the exponent is divisible by $4$, this road suffices (Section 3).
Second, Kummer’s road reaches the regular primes——if a prime $p$ is regular ($p$ divides none of the numerators of the Bernoulli numbers $B_2,\ldots,B_{p-3}$), there is no solution with exponent $p$. Of the $167$ primes from $3$ to below $1000$, $103$ are regular (a share of $0.616766$), and the only irregular ones below $100$ are $37$, $59$ and $67$ (Section 4).
Third, and this is the core. Even together, the two roads leave exponents unreached——exponents not divisible by $4$ and with no regular odd prime factor number $\mathbf{92}$ up to $1000$, the smallest being $37$. Each tool reaches a different range of exponents, and only Wiles’s road reached them all (Section 5).
Fourth, the road from the ABC conjecture reaches large exponents (conditionally)——a counterexample would have quality $\log z^n/\log\mathrm{rad}(xyz)$ above $n/3$. For $n\ge5$ that exceeds $5/3$, above the known highest, $1.629912$, checked in Paper 641. None of the $1268$ triples with $c<10^6$ has $q>5/3$. Triples exceeding the lower bounds $1$ and $4/3$ for $n=3$ and $4$, however, do exist (Section 6).
Fifth, in the world of polynomials it takes one line, and near misses are not solutions——an exhaustive search over polynomials with small coefficients found $20$ pairs for $n=2$ and none for $n=3$ or $4$ (a consequence of Mason--Stothers). Among integers there are $18$ triples up to $2000$ that miss by one, such as $9^3+10^3=12^3+1$, but none that match exactly (Sections 2 and 7).
Keywords: Fermat’s Last Theorem, infinite descent, regular primes, Bernoulli numbers, Kummer.
Scope (what is not claimed)
No new theorem or law is claimed——the Last Theorem and its proof (Wiles, and Taylor and Wiles, $1995$), infinite descent for $n=4$ (Fermat), Kummer’s theorem on regular primes (around $1850$), the list of irregular primes, the conjecture that the share of regular primes is near $e^{-1/2}$ (Siegel), and that the ABC conjecture excludes large exponents (the exposition of Granville and Tucker and others) are all known. The Last Theorem is not proved——Wiles’s road (the Frey curve and modularity) cannot be followed in numbers, so only its address is written. It is not called “distinct roots”——what divides here is not the source of a phenomenon but the tool of proof, so it is called “distinct roads”. Theorems with the same name are kept apart——Fermat’s test (the little theorem) in Paper 603 and Kummer’s theorem on carries in Paper 608 are different theorems from the Fermat and Kummer of this paper. Relation to earlier papers: Paper 641 counted that the ABC conjecture is a theorem for polynomials and a conjecture for integers, and placed Fermat for polynomials as a one-line consequence——this paper places that polynomial road and the road from ABC within a roll call of roads to the integer Last Theorem. Paper 31 gave the $\pi$ of Ramanujan’s series the address of a period of modular forms——Wiles’s road also passes through modular forms. What is added is deciding regularity up to $1000$ by two routes, counting the exponents reached by neither descent nor Kummer, matching the lower bound $n/3$ on a counterexample’s quality with known triples, and placing the separator on the range of exponents each tool reaches.
1. Notation and the Question
The Last Theorem: for $n\ge3$, $x^n+y^n=z^n$ has no solution in coprime positive integers. For a divisor $d$ of $n$, a solution of $x^n+y^n=z^n$ gives a solution of $(x^{n/d})^d+(y^{n/d})^d=(z^{n/d})^d$. Every $n\ge3$ is divisible by $4$ or by an odd prime, so it suffices to show there is no solution for $n=4$ and for odd primes $p$.
Question: which tools reach this “no solution”, and up to which exponents?
2. The World of Polynomials
Computation 2.1: from the $124$ polynomials of degree at most $2$ with integer coefficients from $-2$ to $2$, coprime non-constant pairs $x(t)$, $y(t)$ were chosen, and those for which $x^n+y^n$ is the $n$-th power of a polynomial were sought (narrowed by the values at $t=2$ to $8$ and confirmed by factorisation). For $n=2$, $20$ pairs were found (including $(2t)^2+(t^2-1)^2=(t^2+1)^2$); for $n=3$ and $4$, none.
Observation 2.2 in the world of polynomials, every $n\ge3$ is reached in one line. Applying the Mason--Stothers theorem of Paper 641, with $D$ the largest degree, gives $nD\le3D-1$, so $n<3$——but this is the Last Theorem for polynomials, not a road to the Last Theorem for integers. Integers lack the derivative that its proof uses.
3. The Road of Infinite Descent (n = 4)
Computation 3.1: searching $1\le x\le y\le3000$ for an integer $z$ with $x^4+y^4=z^2$ found none. The same search found $534$ coprime solutions of $x^2+y^2=z^2$ (a positive control that the search works).
Observation 3.2 Fermat’s own road denies a solution by showing that any solution would yield a smaller one (infinite descent). Since $x^4+y^4=z^4$ is a special case of $x^4+y^4=(z^2)^2$, this settles $n=4$——and every exponent divisible by $4$ reduces to $n=4$.
4. Kummer’s Road (Regular Primes)
Computation 4.1: an odd prime $p$ is regular when $p$ divides none of the numerators of the Bernoulli numbers $B_2,B_4,\ldots,B_{p-3}$. Regularity was decided by two routes, one using exact Bernoulli numbers and one following only remainders on division by $p$ through a recurrence, and the results agreed for every prime $p<200$. Regularity was then decided up to $p<1000$.
| primes from $3$ to below $1000$ | $167$ |
|---|---|
| regular | $103$ (share $0.616766$) |
| irregular | $64$ |
| irregular primes below $100$ | $37$, $59$, $67$ |
Primes of irregularity index $2$ or more include $157$ ($k=62$, $110$) and $691$ ($k=12$, $200$), $14$ in all; $491$, $617$ and $647$ have index $3$. The share $0.616766$ differs by $+0.010236$ from Siegel’s conjectured $e^{-1/2}=0.606531$.
Observation 4.2 Kummer’s road reaches only the regular primes. And while it is known that there are infinitely many irregular primes, whether there are infinitely many regular primes is still unknown——the very breadth of this road’s reach is an open question.
5. Exponents Neither Road Reaches
Computation 5.1: among the exponents $n=3$ to $1000$, those not divisible by $4$ and with no regular odd prime factor (reducing neither to infinite descent nor to Kummer’s road) number $92$. In increasing order they run $37$, $59$, $67$, $74$, $101$, $103$, $118$, $131$, $134$, $149$. In each, every odd prime factor is irregular, and none is divisible by $4$.
Observation 5.2 this is the core: each tool reaches a different range of exponents, and even together two of them leave $92$ gaps up to $1000$. Infinite descent reaches the multiples of $4$, Kummer’s road the exponents with a regular odd prime factor. An irregular prime such as $37$ itself is reached by neither——the same “no solution” was reached by different roads for different exponents, and only Wiles’s road (the Frey curve and modularity) reached them all.
6. The Road from the ABC Conjecture (Conditional)
Computation 6.1: if coprime $x<y<z$ satisfy $x^n+y^n=z^n$, then $\mathrm{rad}(x^ny^nz^n)=\mathrm{rad}(xyz)\le xyz<z^3$, so the quality of this triple satisfies
$$q=\frac{\log z^n}{\log\mathrm{rad}(xyz)}>\frac{n}{3}.$$$n/3$ is $1.000000$ for $n=3$, $1.333333$ for $4$, $1.666667$ for $5$, $2.000000$ for $6$ and $2.333333$ for $7$.
| known highest quality ($2+3^{10}\times109=23^5$, Paper 641) | $1.629912$ (difference from $5/3$: $-0.036755$) |
|---|---|
| triples with $q>1$ and $c<10^6$ | $1268$ (highest $1.567887$) |
| of which $q>4/3$ | $20$ |
| of which $q>5/3$ | $0$ |
Observation 6.2 for $n\ge5$, a counterexample would be a triple of higher quality than any found so far. If the ABC conjecture is true, only finitely many triples have quality above $5/3$, so counterexamples with large exponents are limited. On the other hand, triples exceeding the lower bounds $1$ and $4/3$ for $n=3$ and $4$ exist even with $c<10^6$, and this road alone cannot exclude them——the road from ABC reaches large exponents, and infinite descent and Kummer’s road take care of the small ones.
7. Near but Not There
Computation 7.1: with $2\le x\le y\le2000$, $18$ triples have $x^3+y^3$ differing from a cube by one ($9$ on the side $z^3+1$ and $9$ on the side $z^3-1$). In increasing order they begin $6^3+8^3=9^3-1$, $9^3+10^3=12^3+1$ ($1729$) and $64^3+94^3=103^3+1$. No triple with $x^3+y^3=z^3$ was found.
Observation 7.2 near misses are no clue to a solution. There are $18$ triples up to $2000$ that miss by one, but none with a miss of $0$——“no solution” is not a matter of closeness but of a structure that cannot be exactly equal.
8. The Separator
Observation 8.1 the separator is the range of exponents each tool reaches. Infinite descent reaches the multiples of $4$, Kummer’s road the regular primes, the road from ABC (conditionally) the large exponents, and the polynomial road the world of polynomials. Each is partial, and $92$ gaps remain up to $1000$——the same “no solution” is supported by different tools over their own ranges, and only Wiles’s road supported all of it.
Observation 8.2 beside Paper 641, Paper 31, Paper 603 and Paper 608. Paper 641 is where the polynomial road and the road from ABC come from, and Paper 31 wrote an address in modular forms. Fermat’s test in Paper 603 and Kummer’s theorem in Paper 608 are different theorems by the same people——the same name, pointing to a different theorem, is a different tool.
Observation 8.3 what is not said. The Last Theorem is not proved, only the address of Wiles’s road is written, and it is not called “distinct roots”——what is shown is up to which exponents each of four roads reaches, and how many exponents they miss.
9. Roll Call and Closing
- no polynomial solutions for $n=3$ and $4$ (Computation 2.1): $20$ pairs for $n=2$.
- the polynomial road is not an integer road (Observation 2.2): no derivative.
- $x^4+y^4=z^2$: none up to $3000$ (Computation 3.1): positive control $534$.
- regular primes: $103$ of $167$ (Computation 4.1): $0.616766$.
- irregular primes below $100$: $37$, $59$, $67$ (Computation 4.1): two routes agree.
- infinitely many regular primes is open (Observation 4.2): the road’s breadth is a question.
- $\mathbf{92}$ exponents reached by neither road (Computation 5.1): the smallest is $37$.
- each tool has its own range (Observation 5.2): the core.
- a counterexample’s quality exceeds $n/3$ (Computation 6.1): beyond the known highest for $n\ge5$.
- $18$ near misses, $0$ exact (Computation 7.1): closeness is no clue.
Closing: To Fermat’s Last Theorem, that $x^n+y^n=z^n$ has no solution for $n\ge3$, infinite descent reaches from $n=4$ and Kummer’s road from the regular primes, and $0.616766$ of the odd primes below $1000$ were regular. Even together the two do not reach $92$ of the exponents up to $1000$ (the smallest being $37$)——since a counterexample’s quality would exceed $n/3$, the road from the ABC conjecture reaches large exponents conditionally, but not the small ones. In the world of polynomials it takes one line, and integer triples that miss by one are not solutions. The separator is the range of exponents each tool reaches. Placed among the earlier papers——the polynomial road and the ABC road of Paper 641 enter the roll call of roads to the integer Last Theorem. To be honest——Wiles’s road is given only an address, and the proof of the Last Theorem is not entered.
References
- Granville, A., & Tucker, T. J. (2002). It’s as easy as abc. Notices of the American Mathematical Society.
- Taylor, R., & Wiles, A. (1995). Ring-theoretic properties of certain Hecke algebras. Annals of Mathematics.
- Washington, L. C. (1997). Introduction to Cyclotomic Fields. Springer.
- Wiles, A. (1995). Modular elliptic curves and Fermat’s Last Theorem. Annals of Mathematics.
- Yamagishi, Y. (2026). The $\pi$ That Carries No Dimension [Paper 31]. Zenodo. 10.5281/zenodo.22090430
- Yamagishi, Y. (2026). 561 Passes Fermat’s Test for Every Base Coprime to It [Paper 603]. Zenodo.
- Yamagishi, Y. (2026). The Odd Numbers in Pascal’s Triangle Are Counted by the Ones in the Row Number Written in Binary [Paper 608]. Zenodo.
- Yamagishi, Y. (2026). The ABC Conjecture Is a Theorem for Polynomials and a Conjecture for Integers: Integers Lack the Derivative [Paper 641]. Zenodo. 10.5281/zenodo.22805478
Reference
Yamagishi, Y. (2026). The Roads to Fermat’s Last Theorem Reach Different Exponents, and Descent and Kummer Together Miss 92 Exponents up to 1000 [Paper 642]. YOI: https://www.yamagishiyuuki.com/yoi/642
DOI: 10.5281/zenodo.22805604
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 642. This paper is a preprint and has not been peer reviewed.