The curve y^2 = X^3 + 17 has rational points Q_1 = (-2, 3), Q_2 = (-1, 4), Q_3 =

Question

The curve y^2 = X^3 + 17 has rational points Q_1 = (-2, 3), Q_2 = (-1, 4), Q_3 = (2, 5), Q_4 = (4, 9), Q_5 = (8, 23). Show that Q_2, Q_4 and Q_5 can be expressed as mQ_1 + nQ_3 for some integers m and n. (addition here is the group law on the curve). Compute Q_6 = -Q_1 + 2Q_3 band Q_7 = 3Q_1 - Q_3. Remember that negation is a reflection across the x-axis. Find the 8th integral point on this curve (you may need a computer). Challenge: Prove that there are only eight integral points on this curve with y > 0.

Explanation / Answer

Given 1. Q2 = ( -1,4) = ( -2m , 3m) + ( 2n,5n) = ( - 2m +2n , 3m+5n)

equating the respective coords : -2m+2n= -1 , 3m+5n= 4 solving the eqns : m= 13/ 16 , n= 5 / 16

2. Q4 = ( 4,9) => -2m+2n = 4 , 3m+5n = 9 solving m = - 1/8 , n= 15 / 8

3. Q5= ( 8,23) => -2m+2n =8 , 3m+5n= 23 solving : m= 3/8 , n= 35/8

4 Q6 = -Q1+ 2Q3= - ( -2,3) + ( 4,!0 ) = (8, 7)

Q7 = 3Q1 -Q3= ( -6 , 9) - ( 2,5) = ( -8 , 4)  

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