Solve in positive integers:
x2+ y2 = (x/y + 17/4)2
The only positive integer solution for (x,y) is (3,4).
I did run a program initially, then did an analytic solution.
----
big = 1000
for x in range(1,big):
for y in range(1,big):
if x**2 + y**2 - (x/y + 17/4)**2 == 0:
print(x,y,x**2 + y**2, (x/y + 17/4)**2)
----
Analytic:
For (x/y + 17/4) to be an integer, x/y must be k + 3/4
x = y*(4k+3)/4
x/y = (4k+3)/4
(x/y + 17/4) = (4k+3+17)/4 = k+5
So substituting for x, we have
[y*(4k+3)/4]^2 + y^2 = (k+5)^2
y^2(k + 3/4)^2 + y^2 = (k+5)^2
y^2(k^2 + 3k/2 + 9/16) + y^2 = (k+5)^2
y^2(k^2 + 3k/2 + 25/16) = (k+5)^2
y^2(16*k^2 + 24*k + 25)/16 = (k+5)^2
y = 4*(k+5)/sqrt(16*k^2 + 24*k + 25)
if k=0, y=4, so first solution is found.
Let f(k) = (16*k^2 + 24*k + 25)
k f(k) sqrt(f(k))
0 25 5
1 65 irrational
2 137 irrational
3 241 irrational
Since the determinant of f(k), (576 - 1600) is negative, so the only value of k making f(k) a perfect square is k=0, f(k)=25
|
|
Posted by Larry
on 2023-02-20 14:17:22 |
Computer and Analytic solution
