The important point is that both results indicate that it is possible to only O(log, n)k) bit operations, where k is a positive integer. (Also, the recent resug a 0 (log, n)k) bit operations.) This...

Extracted text: The important point is that both results indicate that it is possible to only O(log, n)k) bit operations, where k is a positive integer. (Also, the recent resug a 0 (log, n)k) bit operations.) This contrasts strongly with the problem of factori The best algorithm known for factoring an integer requires a number of bit operat . factored, whereas primality testing seems to require only a number of bit operatione less than a polynomial in the number of bits of the integer tested. We capitalize on thi difference by presenting a recently invented cipher system in Chapter 8. 6.2 EXERCISES 1Show that 91 is a pseudoprime to the base 3. 2. Show that 45 is a pseudoprime to the bases 17 and 19. Show that the even integern= 2 (mod n). The integer 161,038 is the smallest even pseudoprime to the base 2. 161,038 = 2 · 73 · 1103 satisfies the congruence 2" = 4. Show that every odd composite integer is a pseudoprime to both the base 1 and the base -1. GEORG FRIEDRICH BERNHARD RIEMANN (1826–1866), the son of a minister, was born in Breselenz, Germany, His elementary education came from his father. After completing his secondary education, he entered Göttingen Uni- versity to study theology. However, he also attended lectures on mathematics. After receiving the approval of his father to concentrate on mathematics, Rie- mann transfered to Berlin University, where he studied under several prominent mathematicians, including Diriot