On the representations of integers by the sextenary quadratic form x2+y2+z2+ 7s2+7t2+ 7u2 and 7-cores
Abstract
In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x2+y2+z2+ 7s2+7t2+ 7u2. We establish the following intriguing inequalities 2b(n)>=a7(n)>=b(n) for n not equal to 0,2,6,16. Here a7(n) is the number of partitions of n that are 7-cores and b(n) is the number of representations of n+2 by the sextenary form (x 2+ y 2+z 2+ 7s 2 + 7t 2+ 7u2)/8 with x,y,z,s,t and u being odd.
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