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Statement

Convention (p. 293). $p(n)$ is the number of partitions of $n$, with $p(0)=1$ and $p(\alpha)=0$ for $\alpha\notin\mathbb N$. So the congruence below holds trivially at every $n$ for which $(m^k\ell^3n+1)/24$ is not a nonnegative integer.

Theorem 1 (p. 294, quoted). "Let $m\ge5$ be prime and let $k$ be a positive integer. A positive proportion of the primes $\ell$ have the property that

$$p\left(\frac{m^k\ell^3n+1}{24}\right)\equiv0\pmod m$$

for every nonnegative integer $n$ coprime to $\ell$."

Example (pp. 294--295 and 303--304). For $m=13$ and $k=1$ the prime $\ell=59$ satisfies the conclusion; taking $n$ in the class $n\equiv1\pmod{24\cdot59}$ gives display (2) of p. 295, $p(59^4\cdot13n+111247)\equiv0\pmod{13}$ for every nonnegative integer $n$. The check that $59$ works is a finite computation with Sturm's theorem (pp. 303--304).

Source. K. Ono, Distribution of the partition function modulo $m$, Ann. of Math. (2) 151 (2000), no. 1, 293--307; Theorem 1 on p. 294, its proof on pp. 300--301. Pages are the journal's, as printed in the running heads of the copy identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page image. The proof (pp. 297--301), with Theorem 6, Proposition 7 and Theorem 8 on which it rests, was read and followed at the level of its steps; the cited results of Serre, Shimura, Cipra and Niwa were taken as stated in the paper and not checked. Nothing here is independently reviewed.

Proof pointer

Pages 297--301. Theorem 6 (p. 297) identifies the generating function $F(m,k;z)=\sum p\big((m^kn+1)/24\big)q^n$ of display (3) (p. 295, summed over $n\ge0$ with $m^kn\equiv-1\pmod{24}$) modulo $m$ with an explicit quotient of a power of $\Delta$ under $U(m^k)$ and $V(24)$ by $\eta^{m^k}(24z)$; Proposition 7 (p. 298) gives $F(m,k+1;z)\equiv F(m,k;z)\mid U(m)\pmod m$. Theorem 8 (p. 299) then places every $F(m,k;z)$ in the reduction modulo $m$ of the space of cusp forms of weight $(m^2-m-1)/2$ on $\Gamma_0(576m)$ with character $\chi\chi_m^{k-1}$, where $\chi$ is the nontrivial quadratic character of conductor $12$ and $\chi_m$ the Kronecker character of $\mathbb Q(\sqrt m)$. In the proof of Theorem 1 (pp. 300--301), the case $F(m,k;z)\equiv0\pmod m$ is immediate for every $\ell$. Otherwise the Shimura lifts of $F(m,k;z)$ lie, modulo $m$, in weight $m^2-m-2$ on $\Gamma_0(576m)$ with trivial character, and a theorem of Serre (stated on p. 300) gives a positive proportion of primes $\ell\equiv-1\pmod{576m}$ at which every form of that space is annihilated by $T_\ell$ modulo $m$; this set $S(m)$ is defined without reference to $k$. For such $\ell$ the commutation of the Shimura correspondence with the Hecke algebra gives $F(m,k;z)\mid T(\ell^2)\equiv0\pmod m$, and the formula (11) for $T(\ell^2)$ (p. 301), applied at $n\ell$ with $\gcd(n,\ell)=1$, kills the middle term because the Legendre symbol of $n\ell$ modulo $\ell$ vanishes, leaving $p\big((m^k\ell^3n+1)/24\big)\equiv0\pmod m$.

Dependencies

Theorem 6, Proposition 7 and Theorem 8 of the same paper; Serre's theorem on Hecke operators modulo $m$ (the paper's [S], 6.4); the Shimura correspondence ([Sh]) in the generality of Cipra and Niwa ([Ci], [Ni]); a lemma of Serre and Stark on $U(m)$ ([S-St], Lemma 1) in the proof of Theorem 8.

Bears on

• Problem 1106: through Corollary 2, every prime $m\ge5$ divides $p(n)$ for some $n\ge1$; the passage from there to the problem's first question is not in the paper and is drawn on the claim page. The theorem says nothing about the second question, whether $F(n)>n$ for all large $n$.