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Mathematics of Computation

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On a certain (mod $2$) identity and a method of proof by expansion

Authors: Richard Blecksmith, John Brillhart and Irving Gerst
Journal: Math. Comp. 56 (1991), 775-794
MSC: Primary 11P83
MathSciNet review: 1068825
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Abstract: We prove the congruence \[ \prod \limits _{\begin {array}{*{20}{c}} {n = 1} \\ {n \nequiv 7\;\pmod {14}} \\ \end {array} }^\infty {(1 - {x^n}) \equiv \sum \limits _{ - \infty }^\infty {({x^{n(3n + 2)}} + {x^{7n(3n + 2) + 2}})\;\pmod 2} } \] by first establishing a related equation, which reduces to the congruence modulo 2. The method of proof (called "expanding zero") is based on a formula of the authors for expanding the product of two triple products. A second proof of the result more fully explicates the various aspects of the method. A parity result for an associated partition function is also included.

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Keywords: Jacobi triple product, quintuple product, <IMG WIDTH="60" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\bmod   2$"> identity, expansion formula
Article copyright: © Copyright 1991 American Mathematical Society