Not Quite an Ellipse
y 2 = x 3 − 3 x + 7
Let x and y be integers satisfying the elliptic curve above.
Find the sum of all distinct values of x .
The answer is 63.
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1 solution
The equation can be written as ( y − 3 ) ( y + 3 ) = ( x + 1 ) 2 ( x − 2 ) which easily yields solutions ( − 1 , 3 ) ( − 1 , − 3 ) ( 2 , 3 ) ( 2 , − 3 ) . When you go to the linked wiki page you will find information about colinear 3 points on a curve. Connecting points (-1,-3) and (2,3) yields point (3,5); connecting (2,-3) and (3,5) yields (59,453).
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Precisely. ;)
OK, we can use the group structure of the elliptic curve to obtain these extra solutions. How do we know there are no more?
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Using Kushal's brute force method we can find there are in fact many more: " for i in range(1000000,1010000): c=math.pow(i,3)-3*i+7 if int(math.sqrt(c))==math.sqrt(c): print (math.sqrt(c),i) " yields 1000002, 1001999, 1002003, 1004002 and five other solutions
Hopefully I am wrong about this. Please let me know how if someone figures it out.
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@Abel McElroy – You have rounding error issues. If x = 1 0 0 0 0 0 2 then x 3 − 3 x + 7 = 1 . 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 4 5 × 1 0 9 = 1 0 0 0 0 0 3 0 0 0 . 0 0 0 0 0 0 0 0 4 5 which is nearly, but not exactly, an integer!
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@Mark Hennings – nice! thank you. Mark has debunked my brute force counter examples. Rounding errors were the root of the problem. Thanks Mark
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@Abel McElroy – I have tested for 1 ≤ x ≤ 2 0 0 0 0 0 0 , so "yes".
The Wikipedia article mentions Siegel's theorem about a finite number of integral points on an elliptic curve. I think that all integral points can be found in the manner of finding a third point on a line passing through two already known points. There are no more integral points on this curve that could be found in this manner. I do not have a proper proof and that is why I posted my incomplete solution as a comment.
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@Maria Kozlowska – There are, of course, only a finite number of integral points. I am less sure that we cannot be sure that there are not some other integral points, which are obtained from other, rational, points by further applications of the group property. Just because we cannot combine the integral points we know about directly to form other integral points does not mean that there aren't others.
I hunted the net quite extensively, and have not been able to find a definitive statement about this elliptic curve...
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@Mark Hennings – I was hoping that all the integral points form a chain of solutions. Now I do not think it is exactly the case.
Does 1 satisfy? Because if x=1 then y 2 = 7 .
I have done this by Python Programming
import math for i in range(1,100): c=math.pow(i,3)-3*i+7 print(math.sqrt(c),i)
I got x = − 1 , 1 , 2 , 3 , 5 9
I am interested to know is there any other mathematical way to solve this