By Loo Keng Hua, Wang Yuan

Owing to the advancements and functions of laptop technological know-how, ma­ thematicians started to take a significant curiosity within the purposes of quantity concept to numerical research approximately two decades in the past. The development completed has been either very important essentially in addition to passable from the theoretical view element. It'or instance, from the 17th century until now, loads of attempt was once made in constructing equipment for approximating unmarried integrals and there have been just a couple of works on a number of quadrature until eventually the 1950's. yet some time past 20 years, a few new equipment were devised of which the quantity theoretic technique is an efficient one. The quantity theoretic procedure should be defined as follows. We use num­ ber idea to build a series of uniformly disbursed units within the s­ dimensional unit dice G , the place s ~ 2. Then we use the series to s lessen a tricky analytic challenge to an mathematics challenge that could be calculated via machine. for instance, we may perhaps use the mathematics suggest of the values of integrand in a given uniformly dispensed set of G to ap­ s proximate the yes quintessential over G such that the valuable order of the s errors time period is proven to be of the absolute best style, if the integrand satis­ fies yes conditions.

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10) J. 11) Then m! mk

G(x) has only one zero 8' in the circle Ixl < aD. 26), the equation g(x) = 0 has no root with modulus I aol. If g(x) = u(x)v(x), where u(x) and vex) are polynomials with integral coefficients and with degrees ~ 1, and if u(8') = 0, then the moduli of the roots of vex) = 0 are all > Iaol. Hence lao! = Ig(O)1 = lu(O)v(O)1 which leads to a contradiction. ~ Iv(O)1 > laol Thus we have the theorem. 5. If g(x) = 0 has only a root 8' with modulus g(x) is irreducible over Q. ~ 1, then Proof. If g(x) = u(x)v(x), where u(x) and vex) are polynomials with integral coefficients and with degrees ~ 1, and if u(8') = 0, then the moduli of the roots of vex) are all < 1.

A,(s)l, l = 0, 1, .. '. Then 81's are all rational integers. Without loss of generality, we may assume that 8 1 > 0, since a, is a PV number. 2. since 8 1 ~ q} + (s - 1) < The generalization of S" 29 sq}. 1. Proof. Io) + O(8;;p) + + O(8;;1-p))(1 + O(8;;1-p))-1 O(8;;1-p). The theorem is proved. It is well-known that 8 1 can be evaluated by Newton's formula and Rematrk. 3) that to take a to be a unit is more advantageous. 2. The generalization of SIC Let; be a number of Q(a) and s Q" = ~ ;=1 ;(i)a(i)".

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