seventh power & squares?

Question

express 9 ^ (7) - 5 ^ (7) into three sets of difference of two squares.

Answer 1

Nu's method will only produce 2 sets of positive integers which satisfy your equation. They are:

9^7-5^7 = 40588^2-40530^2

and

9^7-5^7 = 1176212^2-1176210^2

Of course, we can get 8 solutions total from these by allowing the numbers being squared to be negative or positive, with any possible sign combination.

These will be the only solutions in integers. For if 9^7-5^7=a^2-b^2=(a-b)(a+b), then since 9^7-5^7 is even, either a-b or a+b must be even, so a and b have the same parity. Then in fact BOTH of a-b and a+b are even. We can factor 9^7-5^7 as 2^2*29*40559. By unique factorization, there are only 2 distinct ways to write 2^2*29*40559 as a product of two positive even numbers:
one of the factors is 2*29 and the other is 2*40559, or one of the factors is 2 and the other is 2*29*40559. In the first case, setting a-b=2*29 and a+b=2*40559 gives the first solution above, that Nu gave, and setting a-b=2 and a+b=2*29*40559 gives the second solution above. Allowing a-b and a+b to be negative gives the 8 possible solutions, allowing the numbers being squared to possibly be negative.

Allowing for rational solutions, as you seem to be interested in, we can easily produce infinitely many solutions, just by writing (a-b)=c*2*29/d and (a+b)=d*2*40559/c, and then solving the resulting equations for a and b to find a solution to 9^7-5^7=a^2-b^2 with a and b rational. For instance, setting c=2 and d=3, we get the solution:

9^7-5^7=(365147/6)^2-(364915/6)^2

Answer 2

Here is an easy way to get the solutions:

Factor 9^7 - 5^7 = 4704844 = 2^2 * 29 * 40559.

Now combine the factors in different ways and check if their difference is divisible by two. For instance:

2*40559 and 2*29 differ by even number.

Compute their average:

In the above case it is 40588.

Then compute the difference between the average and the factors.

In the above case it is 40530.

One then obtains:

4704844 = (40588 + 40530)(40588 - 40530) = 40588^2 - 40530^2.

The others should be obtained in a similar manner.