Download Archimedes' Revenge: The Joys and Perils of Mathematics by Paul Hoffman PDF

By Paul Hoffman

An advent to the delights and demanding situations of recent arithmetic.

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1114 111 211 2 a~ I"llS 1173n41 1217 4, 101 0117illtm il n141 1106 917 SI Rn S ln 1in is, wni a. tnt me iMl"1 Wl1 Iiiiiii*1' nn 14 41n Ra Ian n 919 1111, R IlMi t a? a ItS? III? 2 Rn Imt ali 9 UD en -1Cn M3 IlIi ill . a 01nD 33 n a. al B71 1 InS 1 -1- 1. 1 H I4, 4, II 10 4ti1 b 21 ll 1 Mit? a 14,1 141173 D 611 Dl 1-I1 Xli nle a it t-1 " InD iifl 737 41 III? en? nnI li 1 7 u1 IUI~ 41 1 1D2 in lii 41111W I, U C. n 7911lb ' 4*ID 14m III 114 141 D3 4,9 1fl M n 14 35 nn niann n wn7Ina M0 prime Iis71 a.

Maniac II drew a squarespiral diagram for all the primes under 10 million. Sure enough, many uncannily preferred diagonal lines. 42 ARCHIMEDES' REVENGE MANIAC'S DOODLES Euler's formula n2 + n + 41 turns out to be surprisingly good for large values of n. 5 percent of the time. The formula works even better for lower values of n. For values of n under 2,398, there's an even chance of getting a prime. And for values of n under 100, the formula yields 86 primes and only 14 composite numbers. Ulam and his co-workers discovered other primegenerating formulas that are almost as good as Euler's.

In Jeremiah (25:26 and 51:41), the prophet wrote Sheshach for Babel. The second letter of the Hebrew alphabet (b) was replaced by the second-to-last letter (sh), and 48 The Cryptic Case of a Swarthy Stranger 49 the twelfth letter (I) was replaced by the twelfth-to-last letter (ch). ) The cipher is called Athbash, an acronym formed from the first Hebrew letter (a), the last letter (1h), the second letter (b), and the second-to-last letter (sh). The drawback of an elementary substitution cipher is that it can be cracked simply by analyzing the frequency with which each symbol appears.

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