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Linear and Quadratic Square (Posted on 2015-03-15) Difficulty: 3 of 5
Each of A and B is a positive integer with gcd(A,B) =1 such that each of
A2 + 2*B2 and A + 2*B is a perfect square.

What is the smallest value of A + 2*B for which this is possible? Determine the next two smallest values of A+2*B that satisfy the given conditions.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts A way not to do it. | Comment 2 of 4 |

To save people wasting excessive time on dead ends:

x^2=a+2b  
Then b=1/2(x^2-a)
y^2=a^2+2b^2  
So y^2=a^2+2*(1/2(x^2-a))^2  
And 2y^2 = 3a^2-2ax^2+x^4  
Let p=x^2: 2y^2 = 3a^2-2ap+p^2 

This is a quadratic that can be solved for various a, with valid solutions for x, if p is itself a square = x^2.
There is a recurrence: Xn+1 = P Xn + Q Yn + K, Yn+1 = R Xn + S Yn + L. P,Q,R,S are always 3,4,2,3, K and L are both (-2a), and X0 and Y0 both equal a.

So, you might think, just a matter of checking the recurrences for cases where p is square.

The only problem is that it doesn't work. It produces the first solution, a=1, b=12, x=5, y=17, and also many other solutions, none of which satisfy the requirement that a,b are relatively prime. What in fact happens is that as a gets bigger, there are variants of the solution set where XO and YO are different, though all the other values P,Q,R,S,K,L remain correct.

Say, for example, that a=257. Then 2y^2 = 3*257^2-2*257*p+p^2:  x2 - 2 y2 - 514 x + 198147 = 0., with solutions {X0,Y)}, {257,-257}, {257,257}, {-15,-321}, {-15,321}, {-211,-419}, {-211,419} Note that one of these solutions corresponds with Charlie's list; a=257, y=321 is indeed a solution to the given problem. But then p=-15, which is not a square, even though 2*321^2 = 3*257^2+2*257*15+15^2. Working backwards, 321^2=257^2+2*b^2, so b=136. Now we have 257+2*136 =  23^2.

So somehow, the square root of -15 is 23! (In fact, -15 solves for -257+2*136, rather than 257+2*136).

So, though promising-looking, the method doesn't produce helpful results.


  Posted by broll on 2015-03-16 01:18:47
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