• February 13, 2018
• Number Theory By Euler L.

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1)m f (1)xm ) 0 − (f (m) (0) ∓ . . + (−1)m f (0)xm ). Evaluate this formula in the case f (t) = tµ (1 − t)ν , m = µ + ν. ·2·1 xν ν x + . . ·(µ+1) 1 + µ+ν ν! ⎠ ⎝ . ·2·1 xν ν 1 − µ+ν x ± . . ·(ν+1) ν! µ,ν=1,2,... CHAPTER 3 Continued fractions The powerful tool of continued fractions was systematically studied for the ﬁrst time by Huygens in the seventeenth century. These fractions appear in a natural way by means of the Euclidean algorithm and may be used to construct the set of real numbers out of the set of rationals.

2) add Since {kα} − { α} lies in the interval [0, Q up to zero. Setting q = k − we obtain {qα} = {kα} − { α} < 1 . 1) (since q < Q). Now suppose that α is irrational and that there exist only ﬁnitely many solutions pq11 , . . 1). Since α ∈ Q, we can ﬁnd a Q such that α− pj 1 > qj Q for j = 1, . . 3). 4) α− a b with a ∈ Z and b ∈ N. 1) involves q < b. 1). The theorem is proved. • p q The proof is ineﬃcient. Yet we cannot compute best approximations without big computational eﬀort. 2. A ﬁrst irrationality criterion Dirichlet’s approximation theorem leads to a ﬁrst irrationality criterion.

Am ]. 8. 2 prove for n ≥ 2 that qn pn = [an , an−1 , . . , a1 , a0 ] and = [an , an−1 , . . , a2 , a1 ]. 9. 2 show that m (−1)n−1 pm = a0 + . 10. 3. * Prove that the Greedy algorithm applied to n4 with n ∈ N yields a representation as a sum of four Egyptian fractions at most. Show that only in the case n ≡ 1 mod 4 can it happen that the Greedy algorithm returns more than three Egyptian fractions. Is it possible to restrict n further? 12. i) Verify the Erd¨ os–Strauss conjecture for n = 19, 91, 185, 201 by hand calculation.