Goormaghtigh conjecture

The Goormaghtigh conjecture states that the only nontrivial solutions of (xⁿ−1)/(x−1) = (yᵐ−1)/(y−1), with distinct integer bases greater than one and exponents greater than one, yield 31 and 8191.

Connect

Repunit: A number whose base-b representation consists entirely of the digit 1. Each side of the conjecture's equality is a repunit written in a potentially different base.
Mersenne number: A number of the form 2ᵏ−1, where k is a positive integer. Both known values are Mersenne numbers, giving the base-2 side a particularly simple form.
1+2+4+8+16=31: The identity 31=1+2+4+8+16=1+5+25. It is the known nontrivial equality with the smaller common repunit value.
Computational number theory: The use of algorithms and computation to investigate problems in number theory. Finite searches can test many candidate bases and exponents but cannot alone settle an unrestricted conjecture.
Geometric series: A sum whose successive terms have a constant ratio. The repunit formula is the finite geometric-series identity applied to powers of a base.
Multiplicative order: The least positive exponent k for which a given integer is congruent to 1 modulo a coprime modulus. Congruences among the bases and repunit lengths can constrain possible exponents.
1+2+4+8+16+32+64+128+256+512+1024+2048+4096=8191: The identity 8191=1+2+4+⋯+4096=1+90+90². It is the other known nontrivial equality, involving bases 2 and 90.
Pillai's conjecture: The conjecture that the equation xᵃ−yᵇ=1 has only finitely many solutions in integers greater than one. It illustrates a neighboring exponential Diophantine problem where proving finiteness is central.
Diophantine equation: An equation whose solutions are sought among integers or other specified rational numbers. The conjecture asks whether an integer equation has any solutions beyond two known cases.
Congruence relation: An equivalence relation stating that two integers have the same remainder upon division by a modulus. Reducing the equality modulo factors of either repunit yields arithmetic restrictions on the other base.