On the solutions of an equation.

                                                   Kunal A. Borse

     I will start with a little anecdote from the life of mathematician Srinivasa Ramanujan.

Once a friend visited him, while Ramanujan was cooking in his kitchen. His friend was solving a puzzle from the 'Strand' magazine, but he just could not crack it. So he read it out to Ramanujan.

The puzzle goes as follows-    'Find the numbers x and y, given that

                            1+2+3+......+(x-1)+x=x+(x+1)+.....+y and 50<y<500.' 

Much to his friend's amusement, Ramanujan, still busy with his cooking, answered correctly within a moment. "The value of y must be 288 while that of x must be 204,'' said Ramanujan, leaving his friend surprised. This shows how much Ramanujan was 'into' numbers and thought about numbers all the time.

       Now let us look at this problem and try to find more such pairs of numbers, because this sure looks like a very interesting property. Simplifying the above equation, we get

                                                        x^2=y(y+1)/2   or   2x^2-y^2=y

which looks like a modified form of the Pell's equation. But I tried solving it by another method.

By simple observation, we conclude that x=6,y=8 and x=35,y=49 are two solutions.(we exclude the trivial solution x=y=1.)

We can definitely find the other solutions using a computer program involving nested loops which I leave as an exercise to the reader. On doing the same, one will observe that the number of solutions that we are able to find using a computer program is not large.
 I am now giving the list of solutions I have found out using another simple observation.

1. x=6       y=8
2. x=35     y=49
3. x=204   y=288
4. x=1189   y=1681
5. x=6930  y=9800
6. x=40391  y=57121
7. x=235416   y=332928
8. x=1372105   y=1940449
9. x=7997214   y=11309768
10. x=46611179  y=65918161
11. x=271669860  y=384199200
12. x=1583407981  y=2239277041    and so on.( check for yourself!!)

I know that you did not read any of those, but the point is that I found up to 14 digit solutions of the equation. It is quite evident that it is hard to find such large solutions using the limited computational capacity available.
The simple observation that I used to find these solutions was that-the consecutive solutions are almost in a geometric progression with a common difference (3+2^(1.5))=5.82842712....
You can cross-check this fact with the help of a calculator.
However I was unable to prove this fact, you may try and share the same with me.

The equation 2x^2-y^2-y=0 represents a hyperbola and this fact may be useful in dealing with this equation.

I also tried to generalise the idea, for sums of powers of natural numbers. Though it must be noted that dealing with sums of powers can be really tricky ( remember the Fermat's last theorem and the  Euler's conjecture).
It is easy to find the corresponding equations for the sum of squares and sum of cubes,
The equation for the sum of squares is -
    1+4+9+....+x^2=x^2+......+y^2   which gives us 
       2x(2x^2+1)=y(y+1)(2y+1)    ( using the expression for sum of squares of first n natural numbers.)
But checking by a computer program, they do not have any solutions up to 1000000.(excluding the trivial solution x=y=1)
Though this does not prove that solutions do not exist, it tells us that there is little possibility that they do.
Do let me know if you have any interesting views or information on this matter.