an elliptic curve

an elliptic curve by Thomas McClure I

Introduction

[y/root(x)]^2 = (x+1)*(x-1), an elliptic curve Hence, sending and receiving coded messages is somewhat like solving for roots on an elliptic curve, when x is equal to pi, and P and Q are like twin primes, or like a large prime value of pi. y^2 = x*[x2 – 1] = x^3 – x = x^3 – d^2 * x When d is a whole number, such as d = 1, then d is an area of a right triangle with rational sides , and also “…finding whole numbers of d that are [such areas] is equivalent to finding rational points … on elliptic curves.” (p. 192) [K. Devlin. The Millennium Problems, c. 2002] II

Restatement

An elliptic curve has the form of y^2 = x^3 + a*x + b. Take the example of b=0, a=-d^2, where d=1.(p. 192) This equation is the area of a right triangle with rational sides with the problem of finding the whole number d. "[It] is equivalent to the problem of finding rational points ... on certain elliptic curves." (p. 192)

y^2 = x^3 - x, where y is imaginary if x^3 - x is negative. Hence, y is real, if x^3 - x is positive. **Solutions are (0,0), (1,0), (-1,0), and undefined at infinity. (p. 202) That is, in Fig. 6.3, the graph of the single curve is split into two pieces. (p. 193) III

Conclusion

Hence, sending and receiving coded messages is somewhat like solving for roots on an elliptic curve, when x is equal to pi, and P and Q are like twin primes, or like a large prime value of pi.

Appendix “314,159” is a prime number. (p. 106) Snell found pi between 3.14022 and 3.14160. (p. 38) Hence, pairing a prime number 314,159 with a twin prime is with either 314,157 or 314,161. 314,157 is divided by 3 as 104,719, so it is not a prime. Thus 314,161 is a twin prime to 314,159, and the even composite number 314,160 is x, in f*f = (x + 1) * (x – 1), both twin primes. Root(x) is 560; f is root twin primes product. f= (314,160). f/100,000 = 355*/113*.999999 f= (y/root(x)), where y = f*root(f) = square root[f^3] y^2 = f^3 ; f/100,000 = 3.14160; y/100,000 > not = pi*root(pi) y/100,000 = 5.56834752860 Thus the solution of the elliptic equation is a product of a root of pi and pi approximately. 31 415926 535897 932384 626433 832795 028841 = prime http://www.alpertron.com.ar/ECM.HTM 31 415926 535897 932384 626433 832795 028840 =

2 ^ 3 x 3 x 5 x 7 x 19 x 8 581666 150511 x 229 374624 355487 716489