Question

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 $n.$

Answer

Answer: 144

Concepts:estimation exponent whole number operations

Difficulty rating: 2090

Small Hint:

Estimate the fifth root to narrow the possible integer values of $n$

Big Hint:

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

Solution:

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 $144.$

1989 AIME Problem 9

Answer: 144

Difficulty rating: 2090

Small Hint:

Big Hint:

Solution:

Problem 9 in Other Years