OK Thanks again

I just realize that there are 2 points with x = 17:
7 * G = (17,9)
and
24*g = (17,20)
And for the exemple in http://www.royalforkblog.com/2014/09/04/ecc/
the point I was looking for is (17,9)!!!

My goal is to verify a signature in a bitcoin transaction.
I need a public Key which is a point on the elliptic curve: pubKeyPoint = 
privateKey * G = (X, Y)
where G is the generator point.
In a transaction we can find X and Y but sometimes only X.
Same for the ramdom number. In order to sign a transaction
the user pick a random number RN and compute RN * G = (RNx, RNy).
Only RNx is given in a transaction. In order to verify the validity of a 
signature,
I must find Y and RNy. I don't see other way to do it.
In this case p is very large: is there a non-negligible risk of collision
like (17,9) and (17,20) ?

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