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Rational Points on an Elliptic Curve

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(a) Verify that 18^3 - 1^3 = 17 â?¢ 7^3 and find a point on the curve x^3 + y^3 = 17 with rational coordinates.

(b) Make use of the point in (a) above to find a point on the curve X^3 + y^3 = 17 with positive rational coordinates.

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Solution Summary

In this solution we verify that a given rational point lies on a given elliptic curve and find another such point.

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(a) We have:

18^3 - 1^3
= (18 - 1)(18^2 + 18 + 1)
= 17(324 + 17 + 1)
= 17(343)
= 17 * 7^3.

Thus we have:

(18/7)^3 + (-1/7)^3 = 17,

whence the rational point P = (18/7, -1/7) lies on the curve E: x^3 + y^3 = 17.

(b) Since E is an elliptic curve, we can try computing multiples of P until we find one with both coordinates positive. The ...

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