Baker's explicit abc-conjecture and the unconditional resolution of its conditional Diophantine applications

Baker's explicit abc-conjecture (2004) asserts that pairwise coprime positive integers with $a+b=c$ and $c>2$ satisfy $c<\frac{6}{5}N(\log N)^{\omega}/\omega!$, where $N$ is the radical of $abc$ and $\omega$ the number of distinct prime factors of $N$. Laishram and Shorey (2012) showed that this single explicit inequality, assumed as a hypothesis, conditionally resolves a remarkable list of classical Diophantine problems: it reduces the Tijdeman-Zagier (Beal) conjecture for $x^p+y^q=z^r$ with $p,q,r\ge3$ to the exceptional signature families $Q=\{[3,5,p]:7\le p\le23,\ p\text{ prime}\}\cup\{[3,4,p]:p\text{ prime}\}$, eliminates all non-exceptional solutions of the Nagell-Ljunggren equation, gives effective finiteness for the Goormaghtigh equation, and bounds perfect-power values of products of terms in arithmetic progression. The problem is to remove this conditionality: prove or refute Baker's explicit abc-conjecture unconditionally, and establish the unconditional versions of these theorems.

Background

For a positive integer $n$, write $N(n)$ for its radical (the product of its distinct prime divisors, with $N(1)=1$) and $\omega(n)$ for the number of distinct prime divisors. The $abc$ conjecture of Oesterle and Masser (stated as Conjecture 1.1 of the source paper) predicts that for every $\varepsilon>0$ there is a constant $c_\varepsilon$ such that every triple of pairwise coprime positive integers with $a+b=c$ satisfies $c<c_\varepsilon N(abc)^{1+\varepsilon}$. It is among the most consequential open problems in number theory: an effective version would convert a large family of merely qualitative finiteness statements about Diophantine equations into explicit, computable ones. The best unconditional bounds of this type (Stewart and Yu) are only log-exponential in the radical, so nothing close to exponent $1+\varepsilon$ is currently provable. Guided by numerical experiments on abc-triples, Alan Baker proposed a fully explicit form of the conjecture: pairwise coprime $a+b=c$ should satisfy $c<\frac{6}{5}N(\log N)^{\omega}/\omega!$ with $N=N(abc)$ and $\omega=\omega(N)$, the nondegenerate case being $c>2$ (the inequality fails for the degenerate triple $a=b=1$); the factor $(\log N)^{\omega}/\omega!$ is heuristically near the optimal shape (refinement heuristics of Robert, Stewart and Tenenbaum), but no unconditional result of remotely this strength is known. Laishram and Shorey adopted Baker's inequality as their Conjecture 1.2 and proved that, assumed as a hypothesis, it settles a striking list of classical problems, each of which remains open unconditionally.

The generalized Fermat equation (also called the Fermat-Catalan equation) is $x^p+y^q=z^r$ in integers with fixed signature $(p,q,r)$; a solution is non-trivial when $xyz\neq0$ and primitive when $x,y,z$ are coprime. The case $p=q=r$ is Fermat's Last Theorem, proved by Wiles. The Tijdeman-Zagier conjecture (Conjecture 1.3 of the source paper, also known as Beal's conjecture) asserts that there are no non-trivial primitive solutions with $p,q,r\ge3$; a finiteness theorem of Darmon and Granville guarantees at most finitely many primitive solutions for each signature with $1/p+1/q+1/r<1$, but the full conjecture is open. Many individual signatures are settled unconditionally - for example $x^3+y^4=z^5$ (Siksek and Stoll) and $x^2+y^3=z^7$ (Poonen, Schaefer and Stoll), as well as infinite partial families such as $x^3+y^3=z^p$ for a set of prime exponents $p$ of Dirichlet density about $0.63$ (Chen and Siksek) - typically by the modular method: attaching an auxiliary elliptic curve (a Frey curve) to a putative solution and combining modularity of elliptic curves with level lowering (restricting the modular forms that can give rise to the associated Galois representation), often together with descent and Chabauty-type computations (a technique for determining rational points on curves of genus at least 2 using arithmetic in their Jacobians). Assuming Conjecture 1.2, Laishram-Shorey reduced the Tijdeman-Zagier conjecture to the exceptional families $Q=\{[3,5,p]:7\le p\le23,\ p\text{ prime}\}\cup\{[3,4,p]:p\text{ prime}\}$, where $[\cdot,\cdot,\cdot]$ denotes all permutations of a triple, with any solution in $Q$ satisfying $\max(x^p,y^q,z^r)<e^{1758.3353}$; unconditionally, the families in $Q$ - notably the infinite family $[3,4,p]$ over all primes $p$ - are not resolved in general (individual members may be settled).

The Nagell-Ljunggren equation $(x^n-1)/(x-1)=y^q$ with $x,y>1$, $n>2$, $q>1$ asks when a repunit (a number whose base-$x$ expansion consists of $n$ ones) is a perfect power; exactly three 'exceptional' solutions are known: $11^2=(3^5-1)/(3-1)$, $20^2=(7^4-1)/(7-1)$ and $7^3=(18^3-1)/(18-1)$, and the conjecture that there are no others is open; Bennett and Levin proved unconditionally that the parameter $n$ of any further solution has at most three prime factors, counted with multiplicity. The Goormaghtigh equation $(x^m-1)/(x-1)=(y^n-1)/(y-1)$ with $x\ne y$ and $m,n>2$ asks for integers with two non-trivial repunit representations; the only known solutions are $31$ and $8191$, and the conjecture that there are no others is open (computations confirm no further 'Goormaghtigh primes' below $10^{700}$). Finally, consider the product of $k\ge4$ terms of an arithmetic progression, $n(n+d)\cdots(n+(k-1)d)=by^{\ell}$ with $\gcd(n,d)=1$ and all prime factors of $b$ at most $k$ (write $P(b)$ for the largest prime factor of $b$): for $d=1$ the Erdos-Selfridge theorem proves such a product is never a perfect power, but for $d>1$ the case is open. Conditionally on Conjecture 1.2, Laishram-Shorey proved: no non-exceptional Nagell-Ljunggren solutions (Theorem 3); for the Goormaghtigh equation $m\le6$ with $n$ in an explicit restricted range and an effectively computable absolute constant $C$ with $\max(x,y,n)\le C$ (Theorem 5); and for the progression equation $\ell\le7$ with $k<e^{13006.2}$ when $\ell=7$ (Theorem 2). Every one of these conclusions is conditional on Baker's explicit abc-conjecture, whose unconditional truth is the paper's central open problem; the principal scientific residue of the work is the unconditional status of all of the above.

Problem Statement

The problem has one master target and a set of derived targets subsumed by it; together they carry the single scientific objective of removing the dependence of the cited conditional theorems on Baker's explicit abc-conjecture.

(A) Master target - Baker's explicit abc-conjecture (Conjecture 1.2 of Laishram-Shorey, from Baker 2004). Prove, or refute by an explicit counterexample, the statement: all pairwise coprime positive integers $a,b,c$ with $a+b=c$ and $c>2$ (excluding only the degenerate triple $a=b=1$) satisfy

$$c<\frac{6}{5}\,N\,\frac{(\log N)^{\omega}}{\omega!},$$

where $N=N(abc)$ is the radical of $abc$ and $\omega=\omega(N)$ is the number of distinct prime divisors of $N$. An unconditional proof of any explicit abc-inequality of comparable strength - one that, like Conjecture 1.2, yields an effective bound $c<N^{1+\varepsilon}$ for every $\varepsilon>0$ above an explicit radical threshold, with effective constants - equally resolves this target provided its explicit constants carry through the arguments of the source paper that derive (B1)-(B4).

(B) Derived targets - the unconditional versions of the conditional theorems of Laishram-Shorey. Establish, without any unproved hypothesis, each of the following (or refute the corresponding conjectural assertion by an explicit witness): (B1) For the generalized Fermat equation $x^p+y^q=z^r$ with $p,q,r\ge3$: there are no non-trivial primitive solutions outside the exceptional families $Q=\{[3,5,p]:7\le p\le23,\ p\text{ prime}\}\cup\{[3,4,p]:p\text{ prime}\}$ ($[\cdot,\cdot,\cdot]$ denoting all permutations), and any solution in $Q$ satisfies $\max(x^p,y^q,z^r)<e^{1758.3353}$. (B2) The Nagell-Ljunggren equation $(x^n-1)/(x-1)=y^q$ with $x>1$, $y>1$, $n>2$, $q>1$ has no solutions other than the three known exceptional ones $11^2=(3^5-1)/(3-1)$, $20^2=(7^4-1)/(7-1)$, $7^3=(18^3-1)/(18-1)$. (B3) The Goormaghtigh equation $(x^m-1)/(x-1)=(y^n-1)/(y-1)$ with $x>1$, $y>1$, $x\ne y$, $m,n>2$ has only the two known solutions ($31$ and $8191$); or, at minimum, the unconditional effective statement that $m\le6$, that $n$ lies in the corresponding explicit restricted range, and that $\max(x,y,n)$ is bounded by an explicit absolute constant. (B4) The equation $n(n+d)\cdots(n+(k-1)d)=by^{\ell}$ in integers $n\ge1$, $d>1$, $k\ge4$, $b\ge1$, $y\ge1$, $\ell>1$ with $\gcd(n,d)=1$ and $P(b)\le k$ (largest prime factor of $b$ at most $k$, with the convention $P(1)=1$) implies $\ell\le7$, with $k<e^{13006.2}$ when $\ell=7$.

Relationship and completion boundary. (A) implies (B1)-(B4) via the arguments of the source paper, after making the effective constants explicit and completing the implied finite verifications; the derived targets may equally be achieved independently, by any method. The problem is completely resolved when (A) is settled (by proof or counterexample) and each of (B1)-(B4) is established unconditionally or refuted by an explicit witness - so that none of Theorems 2-5 remains conditional on an unproved hypothesis. Partial progress - resolving individual signatures or sub-ranges of $p$, sharpening the conditional bounds, or extending computational searches - is valuable but does not complete the objective. No restrictions on method, parameter range, or answer form are imposed beyond the statements above.

References

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