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A level-one modular form with rational Fourier coefficients is a weighted homogeneous polynomial with rational coefficients in the normalized Eisenstein series $E_4,E_6$. Subtract its constant coefficient times a same-weight monomial. The remainder is a cusp form; divide it by ${\Delta }_0=(E_4^3-E_6^2)/1728=q+O(q^2)$. The valence formula for the modular group shows that ${\Delta }_0$ has no interior zeros, so this quotient is holomorphic and has rational coefficients. Induction on weight proves the assertion. Euler's Bernoulli number formula for even zeta values then gives the corresponding rational weighted polynomial in the unnormalized lattice Eisenstein sums $G_4,G_6$.

For $n\ge 1$, the von Staudt-Clausen theorem statesConsequently the positive denominator of the Bernoulli number $B_{2n}$ in lowest terms is exactly the squarefree product of the primes satisfying $p-1\mid 2n$. This gives its denominator without computing its numerator.^[ref]

For $B_4$, the relevant primes are $2,3,5$. Their reciprocals sum to $31/30$, and $B_4=-1/30$, so the displayed expression is $1$. The theorem concerns even positive indices; $B_1=-1/2$ is handled separately. It explains the recurring small prime factors in denominators of formulas for sums of powers.