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Mathematics of Computation

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Products and sums of powers of binomial coefficients mod $ p$ and solutions of certain quaternary Diophantine systems

Author: Richard H. Hudson
Journal: Math. Comp. 43 (1984), 603-613
MSC: Primary 11D09; Secondary 11E20
MathSciNet review: 758208
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Abstract: In this paper we prove that certain products and sums of powers of binomial coefficients modulo $ p = qf + 1$, $ q = {a^2} + {b^2}$, are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition

\begin{displaymath}\begin{array}{*{20}{c}} {16{p^\alpha } = {x^2} + 2q{u^2} + 2q... ...2},\quad x \equiv 4\pmod q,\alpha \geqslant 1.} \\ \end{array} \end{displaymath}

The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field

$\displaystyle K = Q\left( {i\sqrt {2q + 2a\sqrt q } } \right),$

as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field $ Q({e^{2\pi ip/q}})$.

Let the four cosets of the subgroup A of quartic residues be given by $ {c_j} = {2^j}A,j = 0,1,2,3$, and let

$\displaystyle {s_j} = \frac{1}{q}\sum\limits_{t \in {c_j}} {t,\quad j = 0,1,2,3.} $

Let $ {s_m}$ and $ {s_n}$ denote the smallest and next smallest of the $ {s_j}$ respectively. We give new, and unexpectedly simple determinations of $ {\Pi _{k \in {c_n}}}kf!$ and $ {\Pi _{k \in {c_{n + 2}}}}kf!$, in terms of the parameters x in the above partition of $ 16{p^\alpha }$, in the complicated case that arises when the class number of K is $ > 1$ and $ {s_m} \ne {s_n}$.

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Article copyright: © Copyright 1984 American Mathematical Society

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