Question

Find the number of four tuples (a,b,c,d) of positive integers satisfying all three equations `a^3=b^2`, `c^3=d^2`, `c-a=64` ?

Answer 1

`{a=15^2,b=15^3,c=17^2,d=17^3}` and
`{a=6^2,b=6^3,c=10^2,d=10^3}`

Explanation:

Solving

`{(a^3=b^2), (c^3=d^2), (c-a=64) :}`

for `b,c,d` we obtain the feasible (integer positive) solution

`b = a^(3/2), c = 64 + a, d = (64 + a)^(3/2)`

Regarding the solution for `b` then `a = m^2` and
regarding the solution for `d` then `64+a = n^2`

so

`n^2-m^2=64 = 2^6`

or

`(n+m)(n-m) = p cdot q = 2^6`

and

`n+m = p = {2^6,2^5,2^4}`

For system

`{ (n+m=p), (n-m=q) :}`

the feasible solutions are

`{n = 17, m = 15}` and `{n = 10,m = 6}`

then `a={15^2, 6^2}`
`d ={(sqrt (64+15^2))^3=17^3,(sqrt(64+6^2))^3=10^3}`
`c = {64+15^2=17^2,64+6^2=10^2}`
`b = {15^3, 6^3}`

Answer 2

There are two, namely:

`color(blue)(""(15^2, 15^3, 17^2, 17^3)) = color(blue)(""(225, 3375, 289, 4913))`

`color(blue)(""(6^2, 6^3, 10^2, 10^3)) = color(blue)(""(36, 216, 100, 1000))`

Explanation:

Let:

`p = b/a`

Then:

`p^2 = b^2/a^2 = a^3/a^2=a`

`p^3 = b^3/a^3 = b^3/b^2 = b`

Then since `p` is a positive rational number whose square is a positive integer, `p` must be a positive integer.

Similarly, if we let `q = d/c`, then `q` is a positive integer with:

`q^2 = c`

`q^3 = d`

Then:

`2^6 = 64 = c - a = q^2-p^2 = (q-p)(q+p)`

Both `(q-p)` and `(q+p)` are positive integers and `(q-p) < (q+p)`.

So the only possible factorings of `2^6` give us:

`{ ((q-p) = 1), ((q+p) = 64) :}color(white)(XX)` hence `2q=65,color(white)(X) color(red)(cancel(color(black)(q=65/2)))`

`{ ((q-p) = 2), ((q+p) = 32) :}color(white)(XX)` hence `2q=34,color(white)(X) color(blue)(q = 17),color(white)(X) color(blue)(p = 15)`

`{ ((q-p) = 4), ((q+p) = 16) :}color(white)(XX)` hence `2q=20,color(white)(X) color(blue)(q = 10), color(white)(X) color(blue)(p = 6)`

So there are two possible tuples `(a,b,c,d)` satisfying the conditions, namely:

`color(blue)(""(15^2, 15^3, 17^2, 17^3)) = color(blue)(""(225, 3375, 289, 4913))`

`color(blue)(""(6^2, 6^3, 10^2, 10^3)) = color(blue)(""(36, 216, 100, 1000))`