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1) where p , p . . are all the primes less than p. Clearly, P is not divisible by any of p, p , p · · · 2. Therefore, P must also be a prime with P > p. Thus, the original assumption must have been false and there is, in fact, no largest prime number. There is no general formula or pattern for the primes. Their occurrence is seemingly random although they become less frequent as N increases. For large N, the number of primes ≤ N is approximated by N/ ln N. For example, 100/ ln(100) ≈ 22, compared with 25 actual primes <100.

Very often we need to transfer a factor i from a denominator to a numerator. The key result is 1 i − i. 79) Several algebraic manipulations with complex numbers are summarized in the following equations: (x1 + iy1 ) + (x2 + iy2 ) (x1 + x2 ) + i(y1 + y2 ) Geometric representation of multiplication by i. The point iz is obtained by 90◦ counterclockwise rotation of z in the complex plane. 80) 46 Chapter 3 Algebra (x1 + iy1 )(x2 + iy2 ) x1 + iy1 x2 + iy2 (x1 x2 ) + i(x1 y2 + y1 x2 ) + i2 (y1 y2 ) (x1 x2 − y1 y2 ) + i(x1 y2 + y1 x2 ) (x1 + iy1 )(x2 − iy2 ) (x2 + iy2 )(x2 − iy2 ) x1 x2 + y1 y2 x22 + y22 +i x2 y1 − x1 y2 x22 + y22 .

The rotation is accomplished by multiplying x by − 1. More generally, any complex number z can be rotated by 180◦ on the complex plane by multiplying it by − 1, designating the transformation z → − z. Consider now a rotation by just 90◦ , for example, counterclockwise. 7 Imagining i 45 rotation by z → iz (repressing for the moment what i stands for). A second counterclockwise 90◦ rotation then produces the same result as a single 180◦ rotation. 77) which agrees with the previous definition of i in Eqs.