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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 \(M_2(\mathbb Z)\) of \(2\times 2\) matrices, of number-theoretic problems and theorems for the ring \(\mathbb Z\) of integers. Sample theorems: \(x^m+y^m=z^m\) has solutions \(x,y,z\) in \(\mathrm{GL}_2(\mathbb Z)\) if and only if \(m\) is divisible by neither 6 nor 4; given \(m\), every matrix in \(M_2(\mathbb Z)\) 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; partitions
11C20 Matrices, determinants in number theory
11D41 Higher degree equations; Fermat’s equation

Citations:

Zbl 0655.00010