Pillai conjecture

Squares, cubes and higher powers can lie surprisingly close: 9 and 8 differ by 1. Pillai asks whether a chosen positive gap can keep recurring forever among larger and larger perfect powers. For gap 7, 16−9 and 32−25 are two examples.[1] [2]

The values 16 and 9 are squares seven units apart, while 32 and 25 give a different perfect-power pair at the same gap. Can any chosen positive gap recur in infinitely many distinct pairs when both bases and exponents are free? For a fixed positive integer $k$, the target is finitely many distinct value-pairs $(a^m,b^n)$ satisfying

$$a^m-b^n=k.$$

Each base and exponent is at least two: $a,b,m,n\ge2$. Count values, not alternative spellings: $16=2^4=4^2$ is one value. Excluding pairs of two squares gives an equivalent target, because $(a-b)(a+b)=k$ already bounds their number. For gap seven, the factor pair $1\times7$ gives $4^2-3^2=7$.[1] [2]

Gap 1 is settled: 8 and 9 are the only such pair. The question asks for finiteness for every fixed positive gap. Solving an equation with one base or exponent fixed does not handle all the choices at once.[1] [2]

What remains is uniform control while both bases and both exponents vary. Fixed-base or specialized equations between powers permit logarithmic and modular bounds that depend on their restrictions. The cited prime-power method solves a prescribed equation family with fixed prime and squarefree-part conditions; it supplies no bound for every pair at a general fixed gap.[1] [2]

Squares, cubes and other power families create different patterns within the whole numbers. Their close coincidences test whether those patterns can keep overlapping at a chosen distance forever. If the conjecture holds, such near encounters are finite exceptions, even when neither the base nor the exponent is fixed.[1] [2]

The conjecture connects sparse individual power families to the structure of their entire overlapping union. If it holds, choosing any finite distance bound would leave only finitely many distinct pairs at or below that distance: there are only finitely many positive integer gaps to combine. Beyond those exceptions, every pair would be farther apart than the chosen bound.

That consequence explains what “powers pull apart” must mean when a value has several exponent representations. It is conditional on the conjecture and gives no rate or effective way to find the last exception; the restricted results in the cited papers do not establish it.[1] [2]