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Nested radical redux (Posted on 2022-06-04) Difficulty: 3 of 5
Find all positive integers (x,y) that satisfy y=√(x+x√x)

Find all positive integers (x,y) that satisfy y=√(x+x√(x+x√x))

No Solution Yet Submitted by Jer    
Rating: 5.0000 (2 votes)

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Solution Parts A and B, and some OEIS | Comment 5 of 9 |
I ran a brief program to find solutions.
Part A
for n = {2,3,4,...}
(x,y) = ((n^2 - 1)^2 , n*(n^2 - 1) )

(9,6), (64,24), (225,60), ...
x  y  sqrt(x)  sqrt(1+sqrt(x))
0 0.0 0.0 1.0
9 6.0 3.0 2.0
64 24.0 8.0 3.0
225 60.0 15.0 4.0
576 120.0 24.0 5.0
1225 210.0 35.0 6.0
2304 336.0 48.0 7.0

So n = sqrt(1+sqrt(x)) for all positive integer n
(n=1 corresponds to x=0)
x = (n^2 - 1)^2

y = (x + x*(x**.5))**.5
y = ( (n^2 - 1)^2 + (n^2 - 1)^2*((n^2 - 1)) )**.5
y = ( (n^2 - 1)^2 + (n^2 - 1)^3 )**.5
y = ( (n^4 - 2n^2 + 1) + (n^6 - 3n^4 + 3n^2 -1) )**.5
y = ( (n^6 - 2n^4 + n^2 ) )**.5
y = ( n^2(n^4 - 2n^2 + 1 ) )**.5
y = n*(n^2 - 1)
------------
Part B  uses function I called ff(x)   
I found only 3 x,y pairs for part B.
I checked x values up to 10000000
(64,40), (576,264), (9828225,1313565)

x  y  sqrt(x)  sqrt(1+sqrt(x))
0 0.0 0.0 1.0
64 40.0 8.0 3.0
576 264.0 24.0 5.0
9828225 1313565.0 3135.0 56.0

I was not able to define a pattern is here, but it is notable that oeis shows the following sequence as having only 4 values:  [1,3,5,56]
A086340 Numbers n such that 1 + (n-1)*n*(n+1) is a square.   So this must be the pattern.
------------
def f(x):
    """ y=√(x+x√x) """
    return (x + x*(x**.5))**.5

def ff(x):
    """ y=√(x+x√(x+x√x)) """
    return (x + x*(f(x)))**.5

def g(n):
    """  to generate x values for f """
    return (n**2 - 1)**2

def gg(n):
    """  to generate y values for f """
    return n*(n**2 - 1)


  Posted by Larry on 2022-06-04 11:52:15
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