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Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
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"... 59. Another short proof of Ramanujan’s mod 5 partition congruence, and more We present another novel short proof of Ramanujan’s partition congruence p(5n + 4) ≡ 0 (mod 5) in addition to that given by John L. Drost [2], and go on to prove rather more. Ramanujan made the remarkable observation from a ..."
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59. Another short proof of Ramanujan’s mod 5 partition congruence, and more We present another novel short proof of Ramanujan’s partition congruence p(5n + 4) ≡ 0 (mod 5) in addition to that given by John L. Drost [2], and go on to prove rather more. Ramanujan made the remarkable observation from a table of values of p(n), the number of partitions of n, that p(5n+4) is divisible by 5. He observed and conjectured much more, and his conjectures turned out in the main to be correct. He gave a simple proof, based on identities of Euler and Jacobi, of the above conjecture, and his proof is essentially the one reproduced in Hardy and Wright [3] and referred to by Drost. Ramanujan’s proof relies on manipulating power series, and considering coefficients modulo 5. It is my intention to give a proof of a similar sort, using only the identity of Jacobi mentioned above, and which is more transparent than that of Ramanujan. And further, with a little extra work including the use of Jacobi’s tripleproduct identity, we prove remarkable congruences for the partition function due to Atkin and SwinnertonDyer. As is usual, write (q) ∞ = ∏ (1 − q n). Then n≥1 p(n)q n = 1 (q)∞
Evidence for a conjecture of Pandharipande
, 2000
"... In [3], Pandharipande studied the relationship between the enumerative geometry of certain 3folds and the GromovWitten invariants. In some good cases, enumerative invariants (which are manifestly integers) can be expressed as a rational combination ..."
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In [3], Pandharipande studied the relationship between the enumerative geometry of certain 3folds and the GromovWitten invariants. In some good cases, enumerative invariants (which are manifestly integers) can be expressed as a rational combination