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Integers inside surds (Posted on 2019-02-03) Difficulty: 3 of 5
Find all integers N for which the expression

√(N + 70710678√173) - √(N - 70710678√173)

has an integer value.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Easy Peasy | Comment 1 of 2
We have something of the form:
Sqrt(x_1) - Sqrt(x_2) = M    (Eq 1)
Squaring equation 1:
x_1 + x_2 - 2 Sqrt(x1 * x2) = M^2  (Eq 2)
since x_1 + x_2 and M are integers, x1 * x2 must be the square of some integer, P
x_1 * x_2 = P^2 = N^2 - (70710678)^2(173)  (Eq 3)
N^2 - P^2 = (N-P)(N+P) = 173 * 2^2 * 3^6 * 89^2 * 14713^2 (Eq 4)
The expression on the right side of equation 4 is a prime decomposition. 
The number of ways the factors in equation 4 may be split into two piles (without regard to order) is 2 * 3 * 7 * 3 * 3 = 378. 
(Note: With the exception of square numbers K, the expression K = Prod p_m_j ^m_j can be split into two factors in 
W = Prod (m_j +1) ways).
However, we need to guarantee that N+P and N-P have the same parity, since this is needed to make N a whole number. So, one factor of 2 must appear in N+P and one factor of 2 must appear in N-P.
Therefore, there are 2 * 7 * 3 * 3 = 126 possible values for N.
For each of these 126 arrangement of factors into two piles, we assign N+P to the pile with the larger product, and N-P to the pile with the smaller product, then solve for N. 
Here is an EXAMPLE: 
N + P = 173 * 2 * 3^4 * 14713^2 = 6066854613594,  
N - P = 2 * 3^2 * 89^2 = 142578 for which
N = 3033427378086
I spare myself the tedium of listing the remaining 125 values for N.

Edited on February 6, 2019, 7:29 am
  Posted by FrankM on 2019-02-06 07:26:40

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