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Non-commutative number theory. (English) Zbl 0666.10032
Algebraic K-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 445-449 (1989).
The author considers analogues, in the ring $M\sb 2(\Bbb Z)$ of $2\times 2$ matrices, of number-theoretic problems and theorems for the ring $\Bbb Z$ of integers. Sample theorems: $x\sp m+y\sp m=z\sp m$ has solutions $x,y,z$ in $\mathrm{GL}\sb 2(\Bbb Z)$ if and only if $m$ is divisible by neither 6 nor 4; given $m$, every matrix in $M\sb 2(\Bbb Z)$ is a sum of $[(G(m)+9)/2]$ $m$th powers. (Here $G(m)$ is the number that arises in Waring’s problem: every sufficiently large integer is a sum of $G(m)$ positive $m$th powers.) This is an appealing paper, partly expository in nature, and carefully written in a way calculated to entice its readers into the subject. [For the entire collection see Zbl 0655.00010.]

##### MSC:
 11P99 Additive number theory 11C20 Matrices, determinants (number theory) 11D41 Higher degree diophantine equations
##### Keywords:
sums of powers; ring of integral matrices