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Ramanujan results for middle school?

Mathematics Educators Asked on September 5, 2021

Pls I wonder what Ramanujan’s results could be explained to middle school level audience, ie without using integral etc that is up to university curriculum?

For example Ramanujan’s infinite radicals could be explained easily

$$ 3=sqrt {1+2{sqrt {1+3{sqrt {1+cdots }}}}}$$

One Answer

You can try the Rogers–Ramanujan identities:

  • The number of partitions of $n$ in which adjacent parts are at least 2 apart is the same as the number of partitions of $n$ in which each part ends with 1,4,6,9.
  • The number of partitions of $n$ without 1 in which adjacent parts are at least 2 apart is the same as the number of partitions of $n$ in which each part ends with 2,3,7,8.

For example, taking $n=10$:

  • Partitions in which adjacent parts are at least 2 apart: $$ 10 = 10 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 6 + 3 + 1 $$
  • Partitions in which each part ends with 1,4,6,9: $$ 10 = 9 + 1 = 6 + 4 = 6 + 1 + 1 + 1 + 1 = 4 + 4 + 1 + 1 = 4 + 6times 1 = 10 times 1 $$
  • Partitions without 1 in which adjacent parts are at least 2 apart: $$ 10 = 10 = 8 + 2 = 7 + 3 = 6 + 4 $$
  • Partitions in which each part ends with 2,3,5,8: $$ 10 = 8 + 2 = 7 + 3 = 3 + 3 + 2 + 2 = 2 + 2 + 2 + 2 + 2 $$

Correct answer by Yuval Filmus on September 5, 2021

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