88 lines
5.9 KiB
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88 lines
5.9 KiB
HTML
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<h2>Integer-valued polynomials</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 402</h3><span style="width:300px;color:#666;">Published on Saturday, 17th November 2012, 07:00 pm; Solved by 195</span></div>
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It can be shown that the polynomial <var>n</var><sup>4</sup> + 4<var>n</var><sup>3</sup> + 2<var>n</var><sup>2</sup> + 5<var>n</var> is a multiple of 6 for every integer <var>n</var>. It can also be shown that 6 is the largest integer satisfying this property.
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Define M(<var>a</var>, <var>b</var>, <var>c</var>) as the maximum <var>m</var> such that <var>n</var><sup>4</sup> + <var>a</var><var>n</var><sup>3</sup> + <var>b</var><var>n</var><sup>2</sup> + <var>c</var><var>n</var> is a multiple of <var>m</var> for all integers <var>n</var>. For example, M(4, 2, 5) = 6.
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Also, define S(<var>N</var>) as the sum of M(<var>a</var>, <var>b</var>, <var>c</var>) for all 0 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>a</var>, <var>b</var>, <var>c</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>N</var>.
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We can verify that S(10) = 1972 and S(10000) = 2024258331114.
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Let F<sub><var>k</var></sub> be the Fibonacci sequence:<br />
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F<sub>0</sub> = 0, F<sub>1</sub> = 1 and<br />
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F<sub><var>k</var></sub> = F<sub><var>k</var>-1</sub> + F<sub><var>k</var>-2</sub> for <var>k</var> <img src='images/symbol_ge.gif' width='10' height='12' alt='≥' border='0' style='vertical-align:middle;' /> 2.
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<p>
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Find the last 9 digits of Σ S(F<sub><var>k</var></sub>) for 2 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>k</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 1234567890123.
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