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An elliptic function is meromorphic and $\Lambda$-periodic. The argument principle on opposite sides of a period parallelogram cancels, so zeros and poles have equal total multiplicity. The Weierstrass elliptic functionis elliptic. Its half-period values $e_i$ are distinct, and comparing poles and leading coefficients proves ${\wp }^{\prime 2}=4{\prod }_i(\wp -e_i)$. The half-period addition formula givesDifferentiate and use the same formula at $z={\omega }_1/4$ to identify $(e_1-e_2)(e_1-e_3)=(\wp ({\omega }_1/4)-e_1{)}^2$, yielding the required derivative ratio.

For $n=\prod p_i^{e_i}$ odd, define $(a/n)=\prod (a/p_i{)}^{e_i}$. Multiplying quadratic reciprocity for all prime-factor pairs givesfor positive odd coprime $a,n$; the sign exponents add modulo two, proving Jacobi reciprocity from Legendre reciprocity. Since $959=7\cdot 137$, $(5/7)=-1$ and $(5/137)=(2/5)=-1$, so $(5/959)=1$. Nevertheless $5$ is a nonresidue modulo both prime factors, so $x^2\equiv 5(\mathrm{mod}959)$ has no solution.