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  "body": "\n## Statement\n\nLet $n \\ge 0$ and $m \\ge 1$ be integers, and let $J_n$ and $J_{n+m}$ be [[Bessel functions of the first kind|CZCMRDWDYW]].\nThen $J_n$ and $J_{n+m}$ have no common zero in $\\C$ other than $z = 0$.\n\n## Remarks\n\nBourget stated it as a conjecture in the nineteenth century, and Siegel proved it in 1929.\nThe proof is number-theoretic: a common nonzero zero would be a root of a polynomial with rational coefficients, hence algebraic.\nBut Siegel showed that $J_n(\\alpha)$ is transcendental, so nonzero, for every nonzero algebraic $\\alpha$.\nExample: $J_0$ and $J_1$ have no common positive zero, which also follows from $J_0' = -J_1$ and the fact that the [[zeros|M324R146XH]] of $J_0$ are simple.\nThe cases $m = 1$ and $m = 2$ follow from the recurrence relations and the simplicity of nonzero zeros; the general case is the hard part.\n\n## Explanations\n\n### L1 human\n\nTwo Bessel functions of different whole-number orders never vanish at the same nonzero point.\nIn terms of a drum, two different families of vibration modes never share a frequency in this way.\nIt sounds like a simple fact about wiggly curves, but its proof needs deep results about transcendental numbers.\n\n### L2 human\n\nFor $m = 1$, a common zero $z_0 \\ne 0$ of $J_n$ and $J_{n+1}$ would give $J_n'(z_0) = \\frac{n}{z_0} J_n(z_0) - J_{n+1}(z_0) = 0$, a double zero, which is impossible.\nFor $m = 2$ the relation $J_n + J_{n+2} = \\frac{2(n+1)}{z} J_{n+1}$ reduces it to the case $m = 1$.\nFor larger $m$ the recurrence writes $J_{n+m}$ as a combination of $J_n$ and $J_{n+1}$ whose coefficients are polynomials in $1/z$ with rational coefficients (the Lommel polynomials).\nSo a common zero must be a root of one of these polynomials, and it is algebraic.\nSiegel's transcendence theorem for $J_n$ at algebraic points, a starting point of the theory of $E$-functions, then settles it.\n",
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