Generalizing and Implementing Michael Hirschhorn's AMAZING Algorithm for Proving RamanujanType Congruences
By Edinah Gnang and Doron Zeilberger
.pdf
.ps
.tex
(Original English is exclusively published in the Personal Journal of
Shalosh B. Ekhad and Doron Zeilberger and arxiv.org. A
Spanish Translation
by Serafin Ruiz Cabello, will appear in La Gaceta de la RSME, in the
section `El Diablo de Los Numeros' edited by Fernando Chamizo).
First Written: June 27, 2013.
Last Update: June 28, 2013
When Mike Hirschhorn
showed us his lovely gem,
that gives the simplesttodate proof of Ramanujan's famous result that p(11n+6) is divisible by 11,
we realized that his amazing method can be extended, and taught to a computer, and can prove
even deeper identities. We would have done much more if not for the existence of
Silviu Radu's powerful algorithm that handles any Ramanujan type congruence for any modular form
(of a very general type), but it is still nice to know that in order to prove such
simply stated results, that can be explained to a sevenyearold, one does not need the
intimidating edifice of the "web of modularity", that Ramanujan never mastered, and probably would
not have liked.
Added July 29, 2013: See lines 1314, p. 27, of G. H. Hardy's hardtofind classic pamphlet
(mimeographed notes, edited by Marshal Hall, published by the Institute for Advanced Study, 1937)
Mathemtical Works of Ramanujan, where it says: "... there seems to be no such simple proof that
p(11m+6) =0 (mod 11)"
Read
Silviu Radu's message,
that explains how to deduce our newly discovered congruences by using his powerful algorithm.
Added June 28, 2013: Mike Hirschhorn (instantly!) proved the elementary lemma, and kindly allowed us to
post it
here.
Added July 7, 2013: JeanPaul Allouche came up with essentially the same
proof.
Maple Packages
HIRSCHHORN,
a Maple package that generalizes and implements Mike Hirschhorn's amazing algorithm.
BOYLAN,
a Maple package for conjecturing Ramanujantype recurrences for
p_{a}(n), the number of colored integer partitions of n using a colors.
Once conjectured, they are all provable algorithmically using Radu's algorithm that
specifies an N_{0} such that checking the congruence for
1 ≤ n ≤ N_{0} suffices to prove it for all n.
Sample Input and Output for HIRSCHHORN

To see an article containing computerdiscovered statements (and their (rigorous!) computergenerated proofs) of all the seven nontriviallyequivalent
Ramanujantype congruences (including the three original ones p(5n+4)=0 mod 4, p(7n+5)=0 mod 7, p(11n+6)=0 mod 11)
for primes up to 13 and powers of the eta function up to 3,
the input file
yields the
output file

To see an article containing computerdiscovered statements of all the eleven nontriviallyequivalent
Ramanujantype congruences (including the three original ones p(5n+4)=0 mod 4, p(7n+5)=0 mod 7, p(11n+6)=0 mod 11)
for primes up to 23 and powers of the eta function up to 9, (and computergenerated proofs of eight of them),
the input file
yields the
output file

To see a terse version for primes up to 13 and powers of the eta function up to 3,
the input file
yields the
output file

To see a terse version
for primes up to 23 and powers of the eta function up to 9,
the input file
yields the
output file
Sample Input and Output for BOYLAN