Skip to main content
SpringerLink
Log in
SpringerLink
Log in

A New Witness Identity for \(\varvec{11\mid p(11n+6)}\)

  • Conference paper
  • First Online:
Analytic Number Theory, Modular Forms and q-Hypergeometric Series (ALLADI60 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 221))

Included in the following conference series:

Abstract

Let p(n) be the number of partitions of the positive integer n. A new q-series identity is presented which witnesses Ramanujan’s observation that \(11|p(11 n +6)\) for all \(n\ge 0\) at one glance. This identity can be derived in a natural way by applying an algorithm to present subalgebras of the polynomial ring \(\mathbb {K}[z]\) as \(\mathbb {K}[z]\)-modules.

Dedicated to our friend Krishna Alladi on his 60th birthday

The research of Radu was supported by the strategic program “Innovatives OÖ 2010 plus” by the Upper Austrian Government in the frame of project W1214-N15-DK6 of the Austrian Science Fund (FWF). Both authors were supported by grant SFB F50-06 of the Austrian Science Fund (FWF)

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 299.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 379.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 379.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For us a \(\mathbb {K}\)-algebra R is a commutative ring R with 1 which is also a vector space over \(\mathbb {K}\).

  2. 2.

    I.e., every element \(p_0(t)+ p_1(t) f + \dots + p_4(t) f^4\) in the module has uniquely determined coefficients \(p_j(t) \in \mathbb {Q}[t]\).

  3. 3.

    Not knowing about Gale’s paper at that time.

  4. 4.

    An entertaining application of this situation is shown in the Numberphile video “How to order 43 Chicken McNuggets”: www.youtube.com/watch?v=vNTSugyS038

  5. 5.

    Such a number is called Frobenius number. Note that 1, 2, 3, 6, and 7 are the other integers not contained in the monoid.

References

  1. G.E. Andrews, P. Paule, A. Riese, MacMahon’s partition analysis III: the omega package. Euro. J. Comb. 22, 887–904 (2001)

    Article  MATH  Google Scholar 

  2. G.E. Andrews, P. Paule, A. Riese, MacMahon’s partition analysis VI: a new reduction algorithm. Ann. Comb. 5, 251–270 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. D.S. Dummit, R.M. Foote, Abstract Algebra, 3rd edn. (Wiley, New York, 2005)

    Google Scholar 

  4. D. Gale, Subalgebras of an algebra with a single generator are finitely generated. Proc. Am. Math. Soc. 8, 929–930 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Hardy, P. Aiyar, B. Wilson (eds.), Collected Papers of Srinivasa Ramanujan (Cambridge University Press, Cambridge, 1927). Reprint: AMS Chelsea Publishing (2000)

    Google Scholar 

  6. R. Hemmecke Dancing Samba with Ramanujan Partition Congruences. preprint, 14 pages (2016). (To appear in: J. Symb. Comput.)

    Google Scholar 

  7. D. Kapur, K. Madlener, A Completion Procedure for Computing a Canonical Basis for a \(k\) -Subalgebra, Computers and Mathematics (Springer, Cambridge, MA, 1989), pp. 1–11

    Google Scholar 

  8. G. Köhler, Eta Products and Theta Series Identities (Springer, Berlin, 2011)

    Google Scholar 

  9. P. Paule, C.S. Radu, Partition analysis, modular functions, and computer algebra, Recent Trends in Combinatorics, The IMA Volumes in Mathematics (Springer, Berlin, 2016), pp. 511–544

    Google Scholar 

  10. C.S. Radu, An algorithmic approach to Ramanujan-Kolberg identities. J. Symb. Comput. 68, 1–33 (2014)

    MathSciNet  Google Scholar 

  11. S. Ramanujan, Some properties of \(p(n)\), the number of partitions of \(n\). Proc. Camb. Philos. Soc. 19, 207–210 (1919)

    MATH  Google Scholar 

  12. L. Robbiano, M. Sweedler Subalgebra bases, in Commutative Algebra (Salvador 1988). Lecture Notes in Mathematics, vol. 1430. (Springer, Berlin, 1990), pp. 61–87

    Google Scholar 

  13. A. Torstensson, Canonical bases for subalgebras on two generators in the univariate polynomial ring. Beiträge Algebra Geom. 43(2), 565–577 (2002)

    Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous referee for carefully reading the manuscript and for the suggestions to improve it. Special thanks go to Frank Garvan for very careful editing.

Author information

Authors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, A-4040, Linz, Austria

    Peter Paule & Cristian-Silviu Radu

Authors
  1. Peter Paule
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cristian-Silviu Radu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristian-Silviu Radu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, State College, Pennsylvania, USA

    George E. Andrews

  2. Department of Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Paule, P., Radu, CS. (2017). A New Witness Identity for \(\varvec{11\mid p(11n+6)}\) . In: Andrews, G., Garvan, F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham. https://doi.org/10.1007/978-3-319-68376-8_34

Download citation

Publish with us

Policies and ethics

Search

Navigation