it is true and possible?

Question

13 ^ (5) - 8 ^ (5) = c ^ (2) - b ^ (2)
one example of c and b is 169263 and 169262
give five more examples of c and b
and of course rational numbers for the above equation.
believe me there are more then that.

Answer 1

13^5 - 8^5 = 338525, which has the following divisor pairs:

1 * 338525
5 * 67705
11 * 30775
25 * 13541
55 * 6155
275 * 1231

Why is that important? Because in order to have

338525 = c^2 - b^2 = (c - b)(c + b),

We need to have c-b and c+b be one of those six pairs. You've already given one, which corresponds to the first divisor pair.

To use the others, it's a simple system of equations for each one:

c - b = 5
c + b = 67705
=> c=33855, b=33850

and so forth. So the other five are

c=753, b=478
c=3105, b=3050
c=6783, b=6758
c=15393, b=15382
c=33855, b=33850

Essentially the same process can be done with rational numbers. Choose your favorite denominator, multiply both sides of the equation by its square, and repeat the above process.

On example is c = 8135/12, b = 4175/12.

Answer 2

As a rule, to solve
c^2 - b^2 = (c-b)(c+b) = N, you have to factor N into two numbers which are either both even or both odd. So we need to factor the LHS.

N = 13^5 - 8^5 = 338535 = 5 * 5 * 11 * 1231

So the factorizations of n can be written as:

1 * 338535
5 * (5*11*1231) = 5 * 67705
(5*5) * (11*1231) = 25 * 13541
(5*11) * (5*1231) = 55 * 6155
(5*5*11) * 1231 = 275 * 1231
11 * (5*5*1231) = 11 * 30775

Given a factorization: N = X*Y, X>Y, we can find c and d such that: X = c+d and Y=c-d by setting c = (X+Y)/2 and d=(X-Y)/2.

So each of these factorizations gives a solution to your equation.

For example, if X=1231 and Y=275, then c = (X+Y)/2 = 753 and d=(X-Y)/2 = 478.

Answer 3

13 ^ (5) - 8 ^ (5) = c ^ (2) - b ^ (2)
This can also be written as, (13^3)^2 - (8^3)^2= c^2 - b^2
Consequently, c=13^3 & b= 8^3

Answer 4

here you go:
33855 and 33850
15393 and 15382
6783 and 6758
3105 and 3050
753 and 478

of course i could give you negative numbers as well. lol

Answer 5

Huh?

Answer 6

not true