Enigma 1235: Power struggle
From New Scientist #2391, 19th April 2003 [link]
I have just given a class of students an interesting exercise. I wrote on the board the exact form of a particular number. Then I asked the class to calculate the sum of that number with its own reciprocal. They all got the correct whole number answer.
Then I asked the first student to square the original number and add that square to its own reciprocal. I asked the next student to cube the original number and add that cube to its own reciprocal. And so on around the class, increasing the power by one each time.
One of the students’ answers was 123.
What was the next student’s answer after that?