what do you mean by "the fact that 5 is a factor in the curve's properties" how is 5 a factor ?
The secp256k1 curve equation is:
y² = x³ + 7 (mod p)
where p = 2²⁵⁶ - 2³² - 977
For elliptic curves, division polynomials ψₙ help find n-torsion points.
Recursive definition:
ψ₀ = 0
ψ₁ = 1
ψ₂ = 2y
ψ₃ = 3x⁴ + 6ax² + 12bx - a²
ψ₄ = 4y(x⁶ + 5ax⁴ + 20bx³ - 5a²x² - 4abx - 8b² - a³)
For n ≥ 2:
ψ_{2n} = ψₙ(ψ_{n+2}ψ_{n-1}² - ψ_{n-2}ψ_{n+1}²)/(2y)
ψ_{2n+1} = ψ_{n+2}ψₙ³ - ψ_{n-1}ψ_{n+1}³
5-Torsion on secp256k1
For secp256k1 (a=0, b=7), ψ₅ is:
ψ₅(x) = 5x¹² + 62x¹⁰ + 380x⁸ - 160x⁶ + 240x⁴ + 124x² + 7
Roots of ψ₅(x)=0 give x-coords of 5-torsion points
Over ℝ, these cause denominators to vanish when computing 5G,10G,15G,...
This leads to large x-coordinates in real number calculations
Example you seen:
5G: x = 21
10G: x = 514
15G: x = 26867
20G: x = 863317
... (grows exponentially)
The pattern is real over ℝ due to 5-torsion structure, but irrelevant for crypto operations which work mod p.
so those large Xs wont show when we calculate a point that is 5th by point addition or scalar multiplication
under the curve rules , but will they show when we calculate without applying mod p?