By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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**Example text**

9}' ... ,gN which are not all zero and for which hold3. (4) The ball for N = 4: application of Minkowski's theorem to the ball K r in four-dimensional space leads to a proof of a famous number theoretic result of Lagrange. This states Corollary 7: Lagrange's Theorem. ,. Proof. The proof proceeds through several steps. In the first one solves the geometric problem of finding the volume of the ball K r . This is the set of points For dimensions N = 1 and N = 2 one has vol(Kt ; N vol(Kt;N = 2) = 1f'.

X IN with 0 < aq - p - f3 < t:3. •• , ZN E {O, 1, ... , 9}, Zl =1= o. Show that there exists some natural number n so that the numeral representation of 2 n in base 10 begins with Zl ••• ZN. Exercises on Chapter 2 * 4. 35 = Let a, f3 be real numbers, (p, q) E 71.. (p, q) 1, and suppose that la < q 2. Show that there exist integers x, y with Ixl ~ 3 and ! I Ts lax - y - f31 < 1~5 (Chebyshev's theorem). 5. Let a be an irrational and f3 a real number. Show that there exist infinitely many pairs (p, q) E 71..

O:Ln}) = (O:ln - [O:ln], ... ,00Ln - [O:Ln]), n = 1,2, ... in IRL /7LL to arbitrary sequences Wt with elements (w1(n), ... ,wL{n)), n = 1,2, .... If J = [0:1, /3dx ... X [O:L, /3d is some parallelepiped in [0, I[L aligned along the axes and by A{wt, N, J) one counts the number of sequence elements lying in J from among then Wt is said to be uniformly di3tributed in [0, I[L if the quantity A(wjf,J) tends as N -+ 00 to the volume vol{J) = {/31 - 0:1)'" (/3L - o:d of the arbitrarily chosen solid.

### A Selection of Problems in the Theory of Numbers by Waclaw Sierpinski, I. N. Sneddon, M. Stark

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