Thread: Find all integers m?

Find all integers m such that m + 3 and m^2 + 3m + 3 are perfect cubes?

Originally Posted by fardeen_gen

Find all integers m such that m + 3 and m^2 + 3m + 3 are perfect cubes?

let me give you an algorithm
let m+3=a^3
then m=a^3-3
now put values of perfect cubes (its easy) in place of a^3.for each such value you will get an integral value of m.
Do the same for m^2 + 3m + 3
that is m^2 + 3m + 3-a^3=0
solve this and insert different value of a^3.this time every solution may not be integral.

Originally Posted by fardeen_gen

Find all integers m such that m + 3 and m^2 + 3m + 3 are perfect cubes?

i believe you want an m such that it holds simultaneously..

i had a miscalculation a while ago but i found out the m=-2 is one..

Yes, the question requires m which holds good for both. Which method did you use to arrive at m = -2?

Did you find a solution to this? My 4 month old daughter is tired of waiting while I try to crack it!

I have made a couple of observations.

1. It is easy to show that m = -2 is a solution. This comes from the special case when . Solving for m gives and consequently m= -2 or m = 0.

2. If we let then the second equation can be written as which looks very much like the first equation.

What are the integer solutions to ?