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  "body": "\n## Statement\n\nLet $x$, $y$, $a$, $b$ be [[natural numbers|0MWKJ44V5A]] with $x \u003e 1$, $y \u003e 1$, $a \u003e 1$ and $b \u003e 1$.\nIf $x^a - y^b = 1$, then $x = 3$, $a = 2$, $y = 2$ and $b = 3$.\n\n## Remarks\n\nThe result is also called Mihăilescu's theorem: Eugène Catalan conjectured it in 1844, and Preda Mihăilescu proved it in 2002.\nRobert Tijdeman had shown in 1976, with [[Baker's theorem|WMBJSNG57F]] on linear forms in logarithms, that the [[Diophantine equation|08B1RTW7VZ]] has only finitely many solutions.\nMihăilescu's proof reduces to odd prime exponents and then uses the arithmetic of [[cyclotomic fields|NN77VYSGXR]].\nEquivalently: $8$ and $9$ are the only two consecutive integers that are both perfect powers.\nThe hypothesis $a, b \u003e 1$ is needed, since for $b = 1$ every $x^a - 1$ is a valid $y$.\nEuler had settled the special case $x^2 - y^3 = 1$, whose only positive solution is $(3, 2)$.\n\n## Explanations\n\n### L1 human\n\nPerfect powers are numbers such as $4, 8, 9, 16, 25, 27, 32$.\nAmong them, $8$ and $9$ sit right next to each other.\nCatalan guessed that this never happens again, and Mihăilescu proved it about one and a half centuries later.\n\n### L2 human\n\nThe equation $x^a - y^b = 1$ is an exponential Diophantine equation, with unknown exponents.\nTijdeman's finiteness result is typical of Baker's method: it bounds the solutions, but the bound was far too large to search.\nMihăilescu's proof avoids computation and works with units and class groups of cyclotomic fields, a theme shared with work on Fermat's Last Theorem.\nThe more general Pillai conjecture, that each positive integer occurs only finitely often as a difference of perfect powers, remains open.\n",
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