Lucas-Lehmer sequence

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The Lucas-Lehmer sequence is defined by the recurrence

$s_{0}=4;\,$

$s_{n}=s_{n-1}^{2}-2,\quad n\geq 1.\,$

A003010 A Lucas-Lehmer sequence: a(0) = 4; for n > 0, a(n) = a(n-1)^2 - 2.

{4, 14, 194, 37634, 1416317954, 2005956546822746114, 4023861667741036022825635656102100994, 16191462721115671781777559070120513664958590125499158514329308740975788034, ...}

Closed-form formula

The Lucas-Lehmer sequence has the closed-form formula

$a(n)=(2+{\sqrt {3}})^{2^{n}}+(2-{\sqrt {3}})^{2^{n}},\,$

where $\scriptstyle 2+{\sqrt {3}}\,$ = 3.732050807568877... and $\scriptstyle 2-{\sqrt {3}}\,$ = 0.2679491924311227... are the roots of $\scriptstyle x^{2}-4x+1\,=\,0\,$.

A019973: Decimal expansion of $\scriptstyle 2+{\sqrt {3}}\,$, tangent of 75 degrees ($\scriptstyle {\frac {5\pi }{12}}\,$ radians), or of cotangent of 15 degrees ($\scriptstyle {\frac {\pi }{12}}\,$ radians).

{3, 7, 3, 2, 0, 5, 0, 8, 0, 7, 5, 6, 8, 8, 7, 7, 2, 9, 3, 5, 2, 7, 4, 4, 6, 3, 4, 1, 5, 0, 5, 8, 7, 2, 3, 6, 6, 9, 4, 2, 8, 0, 5, 2, 5, 3, 8, 1, 0, 3, 8, 0, 6, 2, 8, 0, 5, 5, ...}

A019913 Decimal expansion of $\scriptstyle 2-{\sqrt {3}}\,$, tangent of 15 degrees ($\scriptstyle {\frac {\pi }{12}}\,$ radians), or of cotangent of 75 degrees ($\scriptstyle {\frac {5\pi }{12}}\,$ radians).

{2, 6, 7, 9, 4, 9, 1, 9, 2, 4, 3, 1, 1, 2, 2, 7, 0, 6, 4, 7, 2, 5, 5, 3, 6, 5, 8, 4, 9, 4, 1, 2, 7, 6, 3, 3, 0, 5, 7, 1, 9, 4, 7, 4, 6, 1, 8, 9, 6, 1, 9, 3, 7, 1, 9, 4, 4, 1, ...}

See also

• Lucas–Lehmer primality test