Recently, I was quite surprised, when I saw that there are six different n-values, matching different b-values in secp256k1:
+-----+---------------------------------------------------------------------+
| b | n |
+-----+---------------------------------------------------------------------+
| 0x1 | 0xfffffffffffffffffffffffffffffffe06f23032560e83e138ea6fc857fb4794 |
| 0x2 | 0x1000000000000000000000000000000014551231950b75fc4402da1712fc9b71f |
| 0x3 | 0xffffffffffffffffffffffffffffffff4c43534ba6c5e3a57918113a87c50283 |
| 0x4 | 0x100000000000000000000000000000000b3bcacb4593a1c5a86e7eec3783af5dd |
| 0x6 | 0x100000000000000000000000000000001f90dcfcda9f17c1ec7159035a804b0cc |
| 0x7 | 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141 |
+-----+---------------------------------------------------------------------+
They are repeating over and over again, as b-value is incremented. However, it also seems they can be connected in pairs, each giving the same sum:
+-----+---------------------------------------------------------------------+
| b | n |
+-----+---------------------------------------------------------------------+
| 0x1 | 0xfffffffffffffffffffffffffffffffe06f23032560e83e138ea6fc857fb4794 |
| 0x6 | 0x100000000000000000000000000000001f90dcfcda9f17c1ec7159035a804b0cc |
| sum | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffdfffff860 |
+-----+---------------------------------------------------------------------+
| 0x2 | 0x1000000000000000000000000000000014551231950b75fc4402da1712fc9b71f |
| 0x7 | 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141 |
| sum | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffdfffff860 |
+-----+---------------------------------------------------------------------+
| 0x3 | 0xffffffffffffffffffffffffffffffff4c43534ba6c5e3a57918113a87c50283 |
| 0x4 | 0x100000000000000000000000000000000b3bcacb4593a1c5a86e7eec3783af5dd |
| sum | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffdfffff860 |
+-----+---------------------------------------------------------------------+
The question is: does it mean that there is some kind of connection between y^2=x^3+7, and for example y^2=x^3+2? Or maybe there is another connection, where points on curves with identical p-value and n-value can be mapped? Does it mean, that if we have b=0x7, where there are "n" points, and for example b=0xc curve also has the same amount of points, then does it mean we can map them 1:1?
According to Hasse's Theorem: the number of points N on an elliptic curve over a finite field Fp <where p is prime> conforms to the following formula:
This means that the number of points N is typically close to p+1, with some wiggle room depending on the size of p indicated by the 2√p term
Because of this theorem if two elliptic curves have the same number of points they may share certain properties. However, just because they have the same number of points doesn’t mean there’s a 1:1 correspondence between all of their points we can only potentially create such a mapping for some subsets of points and this doesn’t apply to all points on the curves.