By Isaac Asimov

ISBN-10: 0671442465

ISBN-13: 9780671442460

Seventeen essays on numbers and quantity concept and the connection of numbers to dimension, the calendar, biology, astronomy, and the earth.

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Doing so requires one to take into account all of the available symmetry in our counting problem. Since we have λi | a2 + (−1)i b2 , so it follows that there exist µ1 , µ2 ∈ Z such that λi µi = a2 + (−1)i b2 . Switching between the λi and the µi is a crucial ingredient in deriving some of the inequalities present in the deﬁnition of Va,b (R). 18). 3. With a little extra work one is able to show that √ √ R ⊂ [−c X, c X] × (0, c X/|a2 − b2 |], where c > 0 is an absolute constant and X = 2ν λ1 λ2 B/ max{a, b}.

15. C. Hooley, On the numbers that are representable as the sum of two cubes. J. reine angew. Math. 314 (1980), 146–173. 16. C. Hooley, The density of rational points on cubic surfaces. Glasgow Math. J. 42 (2000), 225–237. 17. -S. Phil thesis, Oxford 2008. 18. I. , North-Holland Math. Library 4, NorthHolland, Amsterdam, New York and Oxford, 1986. 19. M. Nair, Multiplicative functions of polynomial values in short intervals. Acta Arith. 62 (1992), 257–269. 20. E. Peyre, Hauteurs et nombres de Tamagawa sur les vari´et´es de Fano.

A change of variable then yields the alternative formula I∞ = (λ1 · · · λs )−1/3 [0,∞)s wτ (z13 + · · · + zs3 − ν)dz. Consider the equation t = z13 + · · · + zs3 , which deﬁnes a surface in Rs of codimension 1. The area of this manifold in the quadrant with all zj s positive is Γ( 43 )s Γ( 3s )−1 t 3 −1 . Hence, by the transformation formula and Fubini’s theorem, one conﬁrms that I∞ = (λ1 · · · λs )−1/3 Γ 4 3 s Γ s 3 ∞ −1 0 ts/3−1 wτ (t − ν)dt. 19). The analysis of the prime variables case is deeper because we need a rather wide central interval, for a reason that will become more transparent in due course.

### Asimov on Numbers by Isaac Asimov

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