Pell Mell

Submitted by Billy from Montreal, QC, 3/4/2000. Original answer by Carl Johan Ragnarsson; this article by Allen Stenger.

Show that for any the number can be written as the difference of the square roots of two consecutive integers.

Hint 1

Work out the first few powers and verify the result for them. Do you see any patterns?

Hint 2

Here are the first 5 powers:

first 5 powers of sqrt(2) - 1

If we turn this around we get a nicer pattern:

first 5 powers of 1 - sqrt(2)

Based on these examples, it appears that the result is correct, and in fact the first square root is the square root of a perfect square. Do you see any relationships between the numbers for different powers?

Want another hint? Click here.

Click here for the complete solution.