Abstract
For coprime positive integers $a+b=c$, write $\mathrm{rad}(abc)$ for the product of the distinct primes dividing $abc$, and define the quality $q=\log c/\log\mathrm{rad}(abc)$. The ABC conjecture says that for every $\varepsilon>0$ only finitely many triples have $q>1+\varepsilon$. The same question can be posed for polynomials, where it is a theorem. This paper proves nothing; it counts both and sets them side by side. No new theorem or law is claimed.
First, among integers there are triples with $q>1$——triples with $q>1$ number $6$ with $c<10^2$, $31$ with $10^3$, $120$ with $10^4$, $418$ with $10^5$ and $1268$ with $10^6$. Among the $151895$ coprime triples with $c<10^3$ there are $31$, a share of $0.000204$ (Section 2).
Second, triples far above $1$ are rare——of the $1268$ triples with $c<10^6$, $102$ have $q>1.2$, $11$ have $q>1.4$ and only $2$ have $q>1.5$; the highest is $1+4374=4375$ at $1.567887$. The triple known for the highest quality, $2+3^{10}\times109=23^5$, has $\mathbf{1.629912}$. The family $1+(3^{2^k}-1)=3^{2^k}$ keeps $q>1$ for $k=1$ to $6$, but approaches $1$, from $1.292030$ at $k=2$ to $1.056393$ at $k=6$ (Section 3).
Third, and this is the core. For polynomials, the same inequality never fails——for coprime polynomials $a+b=c$, $\max\deg$ does not exceed the number of distinct roots of $abc$ minus $1$ (the Mason--Stothers theorem). This inequality, the counterpart of $\log c\le\log\mathrm{rad}(abc)$ for integers, never failed in $600$ triples built to have repeated roots, and the family $a=x^n$, $c=(x+1)^n$ reached equality for $n=2$ to $8$ (Section 4).
Fourth, the polynomial proof uses the derivative, and integers have no such derivative——the Wronskian $W=ab'-a'b$ is not $0$ and is divisible $m-1$ times by each root repeated with weight $m$. In all $600$ triples, $W$ was divisible by $abc/\mathrm{rad}(abc)$. This step yields the polynomial inequality. Integers have nothing corresponding to $a'$, and the step does not carry over (Section 5).
Keywords: abc conjecture, radical, Mason-Stothers theorem, polynomials, Wronskian.
Scope (what is not claimed)
No new theorem or law is claimed——the ABC conjecture (Masser and Oesterlé, around $1985$), the theorem for polynomials (Stothers $1981$, Mason $1984$), the triple $2+3^{10}\times109=23^5$ known for the highest quality, and counts of triples with $q>1$ are all known. The ABC conjecture is not proved——only triples up to $c<10^6$ are counted, and counting is not proof. The merits of Mochizuki’s claimed proof ($2012$) are not discussed——it is only noted that it has not been widely accepted in the mathematical community. It is not said that “with a derivative the integer case would be solved”——what is shown is only which step of the polynomial proof does not carry over to integers. Relation to earlier papers: Paper 236 measured the dimension of the solution set of a differential equation with the Wronskian——here the same Wronskian counts the repeated roots of polynomials. Paper 640 counted that the number of ways to write an even number as a sum of two primes carries factors set by division by small primes, and noted that counting is not proof——here too the meeting of an addition $a+b=c$ with the multiplicative structure $\mathrm{rad}(abc)$ is counted. What is added is counting integer triples up to $10^6$ while matching three ways of counting, checking the polynomial inequality and the divisibility of the Wronskian on random triples, and placing the separator on whether there is a derivative.
1. Notation and the Question
Integers: for coprime positive integers $a<b$ and $c=a+b$, let $\mathrm{rad}(abc)$ be the product of the distinct primes dividing $abc$, and define the quality
$$q=\frac{\log c}{\log\mathrm{rad}(abc)}.$$$q>1$ means $c>\mathrm{rad}(abc)$. The ABC conjecture says that for every $\varepsilon>0$ only finitely many triples have $q>1+\varepsilon$.
Polynomials: for coprime polynomials with complex coefficients $a+b=c$ (not all constant), let $n_0(abc)$ be the number of distinct roots of $abc$. The degree corresponds to the integer $\log c$, and $n_0(abc)$ to $\log\mathrm{rad}(abc)$.
Question: why is the same question a theorem for polynomials and still a conjecture for integers? The difference is looked at in numbers.
2. Counting Integer Triples
Computation 2.1: triples were counted by exhaustive search for $c<2000$, by a count that narrows candidates using $\mathrm{rad}(a) \mathrm{rad}(b) \mathrm{rad}(c)<c$ up to $c<10^5$, and by a faster version of the same narrowing up to $c<10^6$; at each stage the two counts gave exactly the same triples.
| range of $c$ | $<10^2$ | $<10^3$ | $<10^4$ | $<10^5$ | $<10^6$ |
|---|---|---|---|---|---|
| triples with $q>1$ | $6$ | $31$ | $120$ | $418$ | $1268$ |
The $6$ triples with $c<100$ are $1+8=9$, $1+48=49$, $1+63=64$, $1+80=81$, $5+27=32$ and $32+49=81$. Of all $1501$ coprime triples with $c<100$, $6$ have $q>1$ (a share of $0.003997$); with $c<10^3$, $31$ of $151895$ (a share of $0.000204$).
Observation 2.2 triples with $q>1$ exist, but they are rare. In every one, one of $a$, $b$, $c$ contains the same prime many times——for $c$ to exceed $\mathrm{rad}(abc)$, primes must repeat.
3. High-Quality Triples
Computation 3.1: the $1268$ triples with $c<10^6$ were ordered by quality.
| triple | $\mathrm{rad}(abc)$ | $q$ |
|---|---|---|
| $1+4374=4375$ | $210$ | $1.567887$ |
| $343+59049=59392$ | $1218$ | $1.547075$ |
| $121+255879=256000$ | $4290$ | $1.488865$ |
| $37+32768=32805$ | $1110$ | $1.482910$ |
| $1+2400=2401$ | $210$ | $1.455673$ |
Triples with $q$ above $1.1$, $1.2$, $1.3$, $1.4$ and $1.5$ number $379$, $102$, $29$, $11$ and $2$.
Computation 3.2: the triple known for the highest quality, $2+3^{10}\times109=23^5$ ($2+6436341=6436343$), was checked. It is coprime, with $\mathrm{rad}(abc)=15042$, $\log c=15.677471$ and $\log\mathrm{rad}(abc)=9.618602$, giving $q=\mathbf{1.629912}$.
Computation 3.3: in $1+(3^{2^k}-1)=3^{2^k}$, $3^{2^k}-1$ is divisible by $2^{k+2}$, so $\mathrm{rad}(abc)$ falls below $c$. Since $3^{2^{k+1}}-1=(3^{2^k}-1)(3^{2^k}+1)$ and the second factor is even, the power of $2$ grows by at least one with each step in $k$, and the family goes on without end. The divisibility was checked for $k=1$ to $6$.
| $k$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
|---|---|---|---|---|---|---|
| $q$ | $1.226294$ | $1.292030$ | $1.235303$ | $1.155623$ | $1.095349$ | $1.056393$ |
Observation 3.4 there are infinitely many triples with $q>1$, but the quality of this family approaches $1$. What the ABC conjecture calls finite is the set with $q>1+\varepsilon$, not the set with $q>1$——the substance of the conjecture is not exceeding $1$ but that exceeding $1$ by a fixed margin cannot go on forever.
4. Polynomials
Computation 4.1: $a$ and $b$ were built from powers of small polynomials so as to have repeated roots, with $c=a+b$, and $600$ coprime triples were made at random. In every triple $\max\deg\le n_0(abc)-1$ held; the smallest value of $(n_0(abc)-1)-\max\deg$ was $1$, and the largest of $\max\deg/(n_0(abc)-1)$ was $0.875000$.
Computation 4.2: with $a=x^n$, $c=(x+1)^n$ and $b=c-a$, $n_0(abc)=n+1$ for $n=2$ to $8$, reaching the equality $\max\deg=n_0(abc)-1$.
Observation 4.3 this is the core: the inequality that fails for integers never fails for polynomials. For integers, $\log c\le\log\mathrm{rad}(abc)$ failed $1268$ times with $c<10^6$. The same inequality for polynomials, $\max\deg\le n_0(abc)-1$, never failed in $600$ triples, reaching equality but never passing it——what remains a conjecture for integers is, for polynomials, a theorem that needs not even the margin $\varepsilon$.
5. The Key Step: the Wronskian
Observation 5.1 the polynomial proof uses the Wronskian
$$W=ab'-a'b=ac'-a'c$$(the two forms agree because $c=a+b$). If $a$ has a root $x_0$ with weight $m$, both $a$ and $a'$ are divisible by $(x-x_0)^{m-1}$, and so is $W$. The same holds for the roots of $b$ from the first form and of $c$ from the second. Being coprime, $W$ is divisible by $abc/\mathrm{rad}(abc)$. On the other hand, $W$ has degree at most $\deg a+\deg b-1$, so
$$\deg(abc)-n_0(abc)\le\deg a+\deg b-1,$$which gives $\deg c\le n_0(abc)-1$ (and likewise for the other two).
Computation 5.2: in all $600$ triples of Computation 4.1, $W$ was not $0$ and was divisible by $abc/\mathrm{rad}(abc)$.
Observation 5.3 integers have nothing corresponding to $a'$. For polynomials, the derivative carries each repeated root into $W$ “one fewer” times, where it can be set against the bound on the degree. For integers, there is at hand no operation that lowers the repetition of a prime by one and carries it into another number in a way that meshes with addition——this is where a step of the polynomial proof does not carry over to integers.
6. Fermat for Polynomials
Computation 6.1: for $n=2$, $(1-t^2)^2+(2t)^2=(1+t^2)^2$ is a solution in coprime polynomials.
Observation 6.2 for $n\ge3$ there is no solution in coprime polynomials. Applying Mason--Stothers to $x^n+y^n=z^n$, with $D$ the largest degree of $x$, $y$, $z$, gives $nD\le n_0(xyz)-1\le3D-1$. With $D\ge1$ this requires $n<3$——Fermat for polynomials follows in one line. Fermat for integers took more than three hundred years, until Wiles’s proof ($1995$).
7. The Separator
Observation 7.1 the separator is whether there is a derivative. Polynomials have one; the Wronskian counts repeated roots and turns the inequality into a theorem. Integers have no derivative in that role; the inequality fails $1268$ times and survives as a conjecture with the margin $\varepsilon$——the same meeting of addition and repetition splits into theorem and conjecture according to whether there is a derivative.
Observation 7.2 beside Paper 236 and Paper 640. In Paper 236 the Wronskian measured the dimension of the solution set of a differential equation. In Paper 640 the number of ways to write an even number as a sum of two primes carried factors set by remainders on division——here the same Wronskian counts the repeated roots of polynomials and measures the meeting of the addition $a+b=c$ with the multiplicative structure $\mathrm{rad}(abc)$. And as in Paper 640, counting on the integer side is not proof.
Observation 7.3 what is not said. The ABC conjecture is not proved, the merits of Mochizuki’s claimed proof are not discussed, and it is not said that “with a derivative the integer case would be solved”——what is shown is the count of integer triples, the checks of the polynomial inequality and the Wronskian, and which step of the polynomial proof does not carry over to integers.
8. Roll Call and Closing
- $1268$ triples with $q>1$ for $c<10^6$ (Computation 2.1): three ways of counting agree.
- $31$ of $151895$ with $c<10^3$ (Computation 2.1): rare.
- $2$ triples with $q>1.5$ for $c<10^6$ (Computation 3.1): the highest is $1.567887$.
- $2+3^{10}\times109=23^5$ at $\mathbf{1.629912}$ (Computation 3.2): the known highest.
- the family’s quality approaches $1$ (Computation 3.3): $1.292030$ to $1.056393$.
- the substance is the margin $\varepsilon$ (Observation 3.4): not exceeding $1$.
- the polynomial inequality holds in $600$ triples (Computation 4.1): with an equality family.
- what fails for integers is a theorem for polynomials (Observation 4.3): the core.
- $W$ divisible by $abc/\mathrm{rad}(abc)$ (Computation 5.2): the key step.
- integers have no $a'$ (Observation 5.3): Paper 236 and Paper 640.
Closing: For coprime $a+b=c$, triples whose quality $q=\log c/\log\mathrm{rad}(abc)$ exceeds $1$ number $1268$ with $c<10^6$, only $2$ exceed $1.5$, and the known highest, $2+3^{10}\times109=23^5$, has $1.629912$. For polynomials, the same inequality $\max\deg\le n_0(abc)-1$ never fails (the Mason--Stothers theorem)——its proof uses that the Wronskian $ab'-a'b$ is divisible by the repeated part of the roots, and integers have no such derivative. The separator is whether there is a derivative. Placed among the earlier papers——the Wronskian that measured the dimension of solutions in Paper 236 here counts repeated roots. To be honest——neither a proof of the ABC conjecture nor the merits of Mochizuki’s claim is entered, and the count reaches only $c<10^6$.
References
- Granville, A., & Tucker, T. J. (2002). It’s as easy as abc. Notices of the American Mathematical Society.
- Mason, R. C. (1984). Diophantine Equations over Function Fields. Cambridge University Press.
- Stothers, W. W. (1981). Polynomial identities and Hauptmoduln. The Quarterly Journal of Mathematics.
- Yamagishi, Y. (2026). One and the Same “= 0” Returns Five Different Kinds of Collection [Paper 236]. Zenodo. 10.5281/zenodo.22129043
- Yamagishi, Y. (2026). The Number of Ways to Write an Even Number as a Sum of Two Primes Differs About Twofold by Whether It Is Divisible by 3 [Paper 640]. Zenodo.
Reference
Yamagishi, Y. (2026). The ABC Conjecture Is a Theorem for Polynomials and a Conjecture for Integers: Integers Lack the Derivative [Paper 641]. YOI: https://www.yamagishiyuuki.com/yoi/641
DOI: 10.5281/zenodo.22805478
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 641. This paper is a preprint and has not been peer reviewed.