Jeśmanowicz' conjecture for primitive Pythagorean triples

Partial progress on Jeśmanowicz' conjecture

This records partial progress only. It does not prove Jeśmanowicz' conjecture, and no counterexample is obtained. The results below concern the reduction of hypothetical exceptional solutions, especially the all-even sector and the residual $X=13$ and $X=17$ branches.

1. Reduction of the all-even sector

For an exceptional solution in the all-even sector, the reduction gives odd, pairwise-coprime exponents

$$X,Y,Z>1,\qquad 3\nmid XY,$$

together with coprime odd integers $0<P<Q$ and a sign $\eta\in{\pm1}$, such that

$$Q^X+\eta P^X=2r^Y,\qquad Q^X-\eta P^X=2^{Y\nu}s_0^Y.$$

The exact $2$-adic valuations of the two factors are $1$ and $Y\nu$.

Let $\ell\mid X$ be prime and set

$$U=Q^{X/\ell},\qquad V=\eta P^{X/\ell}.$$

Then the corresponding cyclotomic quotient satisfies

$$F_\ell(U,V)=\frac{U^\ell-V^\ell}{U-V}=\ell^\varepsilon W^Y,\qquad \varepsilon\in{0,1}.$$

Here $U,V,W$ are pairwise coprime and $U>|V|\ge1$.

For every prime $q\mid Y$, set

$$W_0=W^{Y/q}.$$

Then an exceptional solution produces a nondegenerate solution of

$$F_\ell(U,V)=\ell^\varepsilon W_0^q.$$

Thus the all-even sector is reduced to this restricted strong Fermat-Catalan problem. In particular, a proof that the latter equation has no nondegenerate solutions under these coprimality hypotheses would eliminate the entire all-even sector.

This is a conditional reduction, not an unconditional proof of Jeśmanowicz' conjecture.

2. The $X=13$ branch

Exact finite-field and modular computations eliminate the coefficient-two transition patterns arising in the $X=13$ reduction.

For the coefficient-free transition, the current derivation reduces the remaining candidate to

$$(X,Y,Z)=(13,25,17).$$

A direct calculation modulo $11$ then gives a contradiction. Every $25$-th power in $\mathbb F_{11}$ is $0$, $1$, or $-1$, while

$$x^{13}=x^3\qquad\text{in }\mathbb F_{11}.$$

The relevant cyclotomic quotients have residue $6$ or $2$, neither of which is a $25$-th power modulo $11$.

One earlier reduction used to isolate the triple $(13,25,17)$ has not yet been independently rechecked in full, so this calculation alone is not being claimed as a complete closure of the $X=13$ branch.

At the generalized-Fermat stage, the surviving normalized targets have the forms

$$A^{13}+B^{13}=2C^p,\qquad 13\mid C,\qquad v_2(A+B)=1,$$

or

$$A^{13}+B^{13}=C^p,\qquad 13\mid C.$$

The second equation is covered by Theorem 1.2 of the recent Best-Dahmen-Freitas preprint. That theorem states that the primitive equation

$$x^{13}+y^{13}=z^n,\qquad \gcd(x,y,z)=1,\qquad 13\mid z,$$

has no nontrivial solution for $n\ge2$.

In the present reduction, the unit-coefficient target satisfies

$$A\ne0,\qquad B\ne0,\qquad C\ne0,$$

and

$$\gcd(A,B)=1,\qquad \gcd(AB,C)=1,\qquad 13\mid C.$$

Hence

$$\gcd(A,B,C)=1,$$

so the hypotheses of the Best-Dahmen-Freitas theorem match this target.

Therefore, conditional on importing that currently unrefereed preprint, the unit-coefficient $X=13$ target is eliminated.

The remaining $X=13$ target is

$$A^{13}+B^{13}=2C^p,\qquad 13\mid C,\qquad v_2(A+B)=1.$$

This coefficient-two equation is not covered by the Best-Dahmen-Freitas theorem.

3. The residual $X=17$ branch

Exact modular transition certificates eliminate the coefficient-free and coefficient-two projections in the $X=17$ branch. The remaining case is a unit-coefficient difference system.

The possible exponents are reduced to

$$Z\in{19,23,29,31}$$

and

$$Y\in{23,25,29,31,35,37,41,43,47,49,53,55,59,61},$$

with additional pairing restrictions inherited from the preceding reductions.

Every surviving state satisfies

$$17(Q-E)=U^Y,\qquad \frac{Q+E}{2}=V^Y,$$

together with

$$\gcd(U,V)=1,\qquad UV=B_0,\qquad U=34R,$$

where $E=\eta P$ and $Q>|E|$.

The residual variables also satisfy

$$\frac{|E|}{Q}<\frac{1}{4095}$$

and the stronger bound

$$\frac{|E|}{Q}<Q^{-2(Z-17)/Z}.$$

Equivalently,

$$U^Y-34V^Y=-34E,\qquad U^Y+34V^Y=34Q.$$

This gives the approximation

$$\left|\frac{U}{V}-34^{1/Y}\right|<12V^{-4095/4094}.$$

Thus every remaining exceptional solution must belong to a finite list of exponent pairs and satisfy a highly constrained perfect-power approximation equation.

4. Gaussian factorization and descent

The residual $X=17$ system also admits the Gaussian factorizations

$$\frac{Q+iE}{1+i}=z^Z,\qquad \frac{Q^{17}+iE^{17}}{Q+iE}=w^Z.$$

Analysis of the full Gaussian $Z$-th root forces a nontrivial part of the prime support of $U$ into an outer cofactor $U_D$. More precisely,

$$U_D>C(Y,Z)^{-1}U^{1-17/Z},$$

where

$$C(Y,Z)=\left[2\sqrt{2},Z\left(\frac{4095}{4094\cdot17}\right)^{17/Z}\right]^{1/Y}.$$

For every remaining pair $(Z,Y)$,

$$U_D>1.$$

Every prime $q\mid U_D$ satisfies

$$q\notin{2,17,Z},\qquad q\ge2Z+1,$$

and carries a Gaussian rank-$Z$ phase.

In the split case,

$$q\equiv1\pmod{4Z},$$

while in the inert case,

$$q\equiv-1\pmod{4Z}.$$

These rank conditions alone do not yield a contradiction. Nonprincipal local phase configurations satisfying them can occur.

There is also a diagonal descent. Writing the full Gaussian root as

$$h=M+iN,$$

the diagonal branch gives

$$M-\sigma_ZN=e_0^{17},\qquad M+\sigma_ZN=q_0^{17}.$$

Here

$$\sigma_Z=1\qquad\text{for }Z\in{19,23,31},$$

and

$$\sigma_Z=-1\qquad\text{for }Z=29.$$

The resulting smaller pair satisfies

$$\frac{q_0+e_0}{2}=a_0^Y,\qquad \frac{q_0^{17}+e_0^{17}}{q_0+e_0}=a_1^Y,$$

and

$$17Z(q_0-e_0)=d_0^Y,\qquad \frac{q_0^{17}-e_0^{17}}{17(q_0-e_0)}=d_1^Y.$$

Most importantly, its height contracts strictly:

$$\max{|q_0|,|e_0|}<2^{1/34}Q^{1/Z}<Q.$$

This is not yet an infinite descent, because the smaller pair is not presently known to inherit the Gaussian $Z$-th-power condition needed to repeat the construction.

The complementary cyclotomic factors are simultaneously perfect $17$-th powers of Lehmer type at index $Z$. Primitive-divisor results alone do not finish the argument because a primitive divisor can occur with valuation divisible by $17$.

The rational diagonal branch is excluded for every

$$Z\in{19,23,29,31},$$

and the principal $w$-branch is excluded for

$$Z\in{29,31}.$$

Nonprincipal branches remain.

5. Finiteness of each fixed residual branch

For fixed $(Z,Y)$ and fixed Gaussian branch index $k$, the quotient $N/M$ approximates

$$\tan\left(\frac{-\pi/4+2\pi k}{Z}\right)$$

with error

$$O!\left(Q^{-34(Z-17)/Z}\right),$$

while

$$|M|\asymp Q^{17/Z}.$$

Roth's theorem therefore implies that each fixed branch contains only finitely many exceptional solutions.

This finiteness statement is ineffective: it does not provide an explicit bound and does not show that the finite set is empty.

Current status

The reductions above give:

1. an exact conditional reduction of the all-even sector to a restricted strong Fermat-Catalan equation;
2. substantial finite pruning of the $X=13$ and $X=17$ branches;
3. conditional elimination of the unit-coefficient $X=13$ target using Best-Dahmen-Freitas;
4. reduction of the $X=17$ branch to finitely many exponent pairs and an explicit perfect-power system;
5. additional Gaussian rank restrictions and a strict one-step height contraction;
6. finiteness, by Roth's theorem, for every fixed residual Gaussian branch.

The two principal unresolved pieces in this analysis are the coefficient-two $X=13$ target

$$A^{13}+B^{13}=2C^p,\qquad 13\mid C,\qquad v_2(A+B)=1,$$

and the remaining nonprincipal Gaussian/unit-coefficient branches for $X=17$.

The main unresolved issue in the $X=17$ case is global rather than a single missing congruence. The Gaussian, cyclotomic, perfect-power, and local phase constraints are individually compatible; a further argument is needed to force them to be compatible simultaneously.

References

• Amir Ghadermarzi, On The Exceptional solutions of Jeśmanowicz' conjecture, arXiv:2010.07705.
• Best, Dahmen, and Freitas, On the generalized Fermat equation $x^{13}+y^{13}=z^n$, arXiv:2510.12092. The result from this preprint is used conditionally here because the manuscript is currently unrefereed.