a, b, and x are positive integers such that

sqrt(a) + sqrt(b) = sqrt(x)

How many possible values of x less than or equal to 1000 are there?

I didn't have enough attention span to read the other solutions, but apparently I have enough to write my own.  I wonder how that worked out?

It is a good thing that the question asks for the number of possible values for x rather than the number of possible sums.  Otherwise this would be a lot harder!

First, x can be any perfect square up to 1000.  This adds up to 30 values of x (we must exclude x=1).  a and b can simply be squares of integers that add up to sqrt(x).

However, x can be other numbers too.  For example, if x is 44, then sqrt(x) = 2sqrt(11) = sqrt(11) + sqrt(11).  In general, if sqrt(x) can be written as n * sqrt(y) where n is greater than 1, then it can be written as the sum of sqrt((n-1)^2 * y) + sqrt(y).  In fact, this general case includes times x is a perfect square.

Other than the general case shown above, I don't believe there is any possible value for x.  So rather than counting the number of x that qualify, I think it would be easier to count the x that don't qualify.

What x don't qualify?  x doesn't qualify if it isn't divisible by a perfect square.

I used this site for a list of primes.  x's prime factorization can't have a pair of any of them.  I will count the x's, first all the ones with 1 prime factor, then 2, then 3, etc.  Note all "btwn" is exclusive.

There are 169 primes below 1000, including 1.

94 primes btwn 500 and 2.
65 primes btwn 333 and 3.
43 primes btwn 200 and 5.
30 primes btwn 142 and 7.
19 primes btwn 90 and 11.
15 primes btwn 77 and 13.
9 primes btwn 59 and 17.
7 primes btwn 53 and 19.
5 primes btwn 44 and 23.
1 prime btwn 34 and 29.
This adds to a subtotal of 288 values of x.

36 btwn 3 and 167
22 btwn 5 and 100
16 btwn 7 and 72
9 btwn 11 and 46
5 btwn 13 and 39
3 btwn 17 and 30
1 btwn 19 and 27
15 btwn 5 and 67
11 btwn 7 and 48
5 btwn 11 and 31
3 btwn 13 and 26
1 btwn 17 and 20
5 btwn 7 and 29
2 btwn 11 and 19
134 subtotal

8 btwn 5 and 34
5 btwn 7 and 24
1 btwn 11 and 16
2 btwn 7 and 15
16 subtotal

The total is 607 impossible values of x.
There are 393 possible values of x.

I realize my counting method is ridiculously unclear, but I am confident it works.  I merely counted the number of x below 1000 that didn't have any square factors.  I am not so confident that didn't make errors in the actual counting, since I did it by hand.

Edited on May 5, 2005, 11:38 pm

Edited on May 5, 2005, 11:40 pm

Posted by Tristan on 2005-05-05 23:35:58