Generalizing and Implementing Michael Hirschhorn's AMAZING Algorithm for Proving Ramanujan-Type Congruences

By Edinah Gnang and Doron Zeilberger

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(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 simplest-to-date 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 seven-year-old, 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 13-14, p. 27, of G. H. Hardy's hard-to-find 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: Jean-Paul 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 Ramanujan-type 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 N0 such that checking the congruence for 1 ≤ n ≤ N0 suffices to prove it for all n.

# Sample Input and Output for HIRSCHHORN

• To see an article containing computer-discovered statements (and their (rigorous!) computer-generated proofs) of all the seven non-trivially-equivalent Ramanujan-type 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 computer-discovered statements of all the eleven non-trivially-equivalent Ramanujan-type 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 computer-generated 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

• To see an article containing Ramanujan type congruences for p-a(n) for odd a between 1 and 47, reproducing (empirically) Theorem 1.3 in Matthew Boylan's article (with one error corrected, r=27, l=31 should be removed), plus five first terms of the associated integer sequences {p-a(Pn+r)/P} for the P and (r=a/24 mod P)
the input file yields the output file

• To see a more extensive article with Ramanujan type congruences for p-a(n) for odd a between 1 and 399, and primes ≤ 500, extending (empirically) Theorem 1.3 in Matthew Boylan's article (with one error corrected, r=27, l=31 should be removed), plus five first terms of the associated integer sequences {p-a(Pn+r)/P} for the P and (r=a/24 mod P)
the input file yields the output file
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger