1989 AIME 第 9 题

Question

1960

年代，三位美国数学家证明存在正整数

n

使

133^5+110^5+84^5+27^5=n^5\text{，}

从而推翻了欧拉的一个猜想。求

n

的值。

One of Euler’s conjectures was disproved in the

1960

s by three American mathematicians when they showed there was a positive integer

n

such that

133^5+110^5+84^5+27^5=n^5.

Find the value of

n.

答案

估算五次方根，以缩小

n

的整数取值范围

Estimate the fifth root to narrow the possible integer values of

n

通过反复平方和乘法计算各个五次方，再将它们的和与附近的候选值比较

Evaluate the fifth powers by repeated squaring and multiplication, then compare their sum with the nearby candidate

直接进行整数运算，得到

\begin{aligned}133^5&=41{,}615{,}795{,}893,\\110^5&=16{,}105{,}100{,}000,\\84^5&=4{,}182{,}119{,}424,\\27^5&=14{,}348{,}907\end{aligned}\text{。}

它们的和为

61{,}917{,}364{,}224

。反复相乘也可得

144^5=61{,}917{,}364{,}224

，所以正整数

n

为

144

。

Direct integer arithmetic gives

\begin{aligned}133^5&=41{,}615{,}795{,}893,\\110^5&=16{,}105{,}100{,}000,\\84^5&=4{,}182{,}119{,}424,\\27^5&=14{,}348{,}907.\end{aligned}

Their sum is

61{,}917{,}364{,}224.

Repeated multiplication also gives

144^5=61{,}917{,}364{,}224,

so the positive integer

n

is

144.

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