Does the elliptic curve equation y^{2}= x^{3}- 7x - 6 over real numbers define a group?

Yes, since

4a^{3}+ 27b^{2}= 4(-7)^{3}+ 27(-6)^{2}= -400

The equation y^{2}= x^{3}- 7x - 6 does define an elliptic curve group because 4a^{3}+ 27b^{2}is not 0.

What is the additive identity of regular integers?

The additive identity of regular integers is 0, since x + 0 = x for all integers.

Is (4,7) a point on the elliptic curve y^{2}= x^{3}- 5x + 5 over real numbers?

Yes, since the equation holds true for x = 4 and y = 7:

(7)^{2}= (4)^{3}- 5(4) + 5

49 = 64 - 20 + 5

49 = 49

What are the negatives of the following elliptic curve points over real numbers?

P(-4,-6), Q(17,0), R(3,9), S(0,-4)

The negative is the point reflected through the x-axis. Thus

-P(-4,6), -Q(17,0), -R(3,-9), -S(0,4)

In the elliptic curve group defined by y^{2}= x^{3}- 17x + 16 over real numbers, what is P + Q if P = (0,-4) and Q = (1,0)?

From the Addition formulae:

s = (y_{P}- y_{Q}) / (x_{P}- x_{Q}) = (-4 - 0) / (0 - 1) = 4

x_{R}= s^{2}- x_{P}- x_{Q}= 16 - 0 - 1 = 15

and

y_{R}= -y_{P}+ s(x_{P}- x_{R}) = 4 + 4(0 - 15) = -56

Thus P + Q = (15, -56)

In the elliptic curve group defined by y^{2}= x^{3}- 17x + 16 over real numbers, what is 2P if P = (4, 3.464)?

From the Doubling formulae:

s = (3x_{P}^{2}+ a) / (2y_{P}) = (3*(4)^{2}+ (-17)) / 2*(3.464) = 31 / 6.928 = 4.475

x_{R}= s^{2}- 2x_{P}= (4.475)^{2}- 2(4) = 20.022 - 8 = 12.022

and

y_{R}= -y_{P}+ s(x_{P}- x_{R}) = -3.464 + 4.475(4 - 12.022) = - 3.464 - 35.898 = -39.362

Thus 2P = (12.022, -39.362)

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